# Dokchitser’s L-functions Calculator¶

AUTHORS:

• Tim Dokchitser (2002): original PARI code and algorithm (and the documentation below is based on Dokchitser’s docs).

• William Stein (2006-03-08): Sage interface

Todo

• add more examples from SAGE_EXTCODE/pari/dokchitser that illustrate use with Eisenstein series, number fields, etc.

• plug this code into number fields and modular forms code (elliptic curves are done).

class sage.lfunctions.dokchitser.Dokchitser(conductor, gammaV, weight, eps, poles=None, residues='automatic', prec=53, init=None)

Dokchitser’s $$L$$-functions Calculator

Create a Dokchitser $$L$$-series with

Dokchitser(conductor, gammaV, weight, eps, poles, residues, init, prec)

where

• conductor – integer, the conductor

• gammaV – list of Gamma-factor parameters, e.g. [0] for Riemann zeta, [0,1] for ell.curves, (see examples).

• weight – positive real number, usually an integer e.g. 1 for Riemann zeta, 2 for $$H^1$$ of curves/$$\QQ$$

• eps – complex number; sign in functional equation

• poles – (default: []) list of points where $$L^*(s)$$ has (simple) poles; only poles with $$Re(s)>weight/2$$ should be included

• residues – vector of residues of $$L^*(s)$$ in those poles or set residues=’automatic’ (default value)

• prec – integer (default: 53) number of bits of precision

RIEMANN ZETA FUNCTION:

We compute with the Riemann Zeta function.

sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L
Dokchitser L-series of conductor 1 and weight 1
sage: L(1)
Traceback (most recent call last):
...
ArithmeticError
sage: L(2)
1.64493406684823
sage: L(2, 1.1)
1.64493406684823
sage: L.derivative(2)
-0.937548254315844
sage: h = RR('0.0000000000001')
sage: (zeta(2+h) - zeta(2.))/h
-0.937028232783632
sage: L.taylor_series(2, k=5)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5)


RANK 1 ELLIPTIC CURVE:

We compute with the $$L$$-series of a rank $$1$$ curve.

sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(algorithm='gp'); L
Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L(1)
0.000000000000000
sage: L.derivative(1)
0.305999773834052
sage: L.derivative(1,2)
0.373095594536324
sage: L.num_coeffs()
48
sage: L.taylor_series(1,4)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4)
sage: L.check_functional_equation()  # abs tol 1e-19
6.04442711160669e-18


RANK 2 ELLIPTIC CURVE:

We compute the leading coefficient and Taylor expansion of the $$L$$-series of a rank $$2$$ elliptic curve.

sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.num_coeffs()
156
sage: L.derivative(1,E.rank())
1.51863300057685
sage: L.taylor_series(1,4)
-1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4)  # 32-bit
-2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4)  # 64-bit


NUMBER FIELD:

We compute with the Dedekind zeta function of a number field.

sage: x = var('x')
sage: K = NumberField(x**4 - x**2 - 1,'a')
sage: L = K.zeta_function(algorithm='gp')
sage: L.conductor
400
sage: L.num_coeffs()
264
sage: L(2)
1.10398438736918
sage: L.taylor_series(2,3)
1.10398438736918 - 0.215822638498759*z + 0.279836437522536*z^2 + O(z^3)


RAMANUJAN DELTA L-FUNCTION:

The coefficients are given by Ramanujan’s tau function:

sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)


We redefine the default bound on the coefficients: Deligne’s estimate on tau(n) is better than the default coefgrow(n)=(4n)^{11/2} (by a factor 1024), so re-defining coefgrow() improves efficiency (slightly faster).

sage: L.num_coeffs()
12
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L.num_coeffs()
11


Now we’re ready to evaluate, etc.

sage: L(1)
0.0374412812685155
sage: L(1, 1.1)
0.0374412812685155
sage: L.taylor_series(1,3)
0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3)

check_functional_equation(T=1.2)

Verifies how well numerically the functional equation is satisfied, and also determines the residues if self.poles != [] and residues=’automatic’.

More specifically: for $$T>1$$ (default 1.2), self.check_functional_equation(T) should ideally return 0 (to the current precision).

• if what this function returns does not look like 0 at all, probably the functional equation is wrong (i.e. some of the parameters gammaV, conductor etc., or the coefficients are wrong),

• if checkfeq(T) is to be used, more coefficients have to be generated (approximately T times more), e.g. call cflength(1.3), initLdata(“a(k)”,1.3), checkfeq(1.3)

• T=1 always (!) returns 0, so T has to be away from 1

• default value $$T=1.2$$ seems to give a reasonable balance

• if you don’t have to verify the functional equation or the L-values, call num_coeffs(1) and initLdata(“a(k)”,1), you need slightly less coefficients.

EXAMPLES:

sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()  # abs tol 1e-19
-2.71050543121376e-20


If we choose the sign in functional equation for the $$\zeta$$ function incorrectly, the functional equation doesn’t check out.

sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()
-9.73967861488124

derivative(s, k=1)

Return the $$k$$-th derivative of the $$L$$-series at $$s$$.

Warning

If $$k$$ is greater than the order of vanishing of $$L$$ at $$s$$ you may get nonsense.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.derivative(1,E.rank())
1.51863300057685

gp()

Return the gp interpreter that is used to implement this Dokchitser L-function.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L(2)
0.546048036215014
sage: L.gp()
PARI/GP interpreter

init_coeffs(v, cutoff=1, w=None, pari_precode='', max_imaginary_part=0, max_asymp_coeffs=40)

Set the coefficients $$a_n$$ of the $$L$$-series.

If $$L(s)$$ is not equal to its dual, pass the coefficients of the dual as the second optional argument.

INPUT:

• v – list of complex numbers or string (pari function of k)

• cutoff – real number = 1 (default: 1)

• w – list of complex numbers or string (pari function of k)

• pari_precode – some code to execute in pari before calling initLdata

• max_imaginary_part – (default: 0): redefine if you want to compute L(s) for s having large imaginary part,

• max_asymp_coeffs – (default: 40): at most this many terms are generated in asymptotic series for phi(t) and G(s,t).

EXAMPLES:

sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)


Evaluate the resulting L-function at a point, and compare with the answer that one gets “by definition” (of L-function attached to a modular form):

sage: L(14)
0.998583063162746
sage: a = delta_qexp(1000)
sage: sum(a[n]/float(n)^14 for n in range(1,1000))
0.9985830631627459


Illustrate that one can give a list of complex numbers for v (see trac ticket #10937):

sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:])
sage: L2(14)
0.998583063162746

num_coeffs(T=1)

Return number of coefficients $$a_n$$ that are needed in order to perform most relevant $$L$$-function computations to the desired precision.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.num_coeffs()
26
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.num_coeffs()
568
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4


Verify that num_coeffs works with non-real spectral parameters, e.g. for the L-function of the level 10 Maass form with eigenvalue 2.7341055592527126:

sage: ev = 2.7341055592527126
sage: L = Dokchitser(conductor=10, gammaV=[ev*i, -ev*i],weight=2,eps=1)
sage: L.num_coeffs()
26

set_coeff_growth(coefgrow)

You might have to redefine the coefficient growth function if the $$a_n$$ of the $$L$$-series are not given by the following PARI function:

coefgrow(n) = if(length(Lpoles),
1.5*n^(vecmax(real(Lpoles))-1),
sqrt(4*n)^(weight-1));


INPUT:

• coefgrow – string that evaluates to a PARI function of n that defines a coefgrow function.

EXAMPLES:

sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L(1)
0.0374412812685155

taylor_series(a=0, k=6, var='z')

Return the first $$k$$ terms of the Taylor series expansion of the $$L$$-series about $$a$$.

This is returned as a series in var, where you should view var as equal to $$s-a$$. Thus this function returns the formal power series whose coefficients are $$L^{(n)}(a)/n!$$.

INPUT:

• a – complex number (default: 0); point about which to expand

• k – integer (default: 6), series is $$O($$

• var – string (default: ‘z’), variable of power series

EXAMPLES:

sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.taylor_series(2, 3)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3)
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.taylor_series(1)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6)


We compute a Taylor series where each coefficient is to high precision.

sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(200, algorithm='gp')
sage: L.taylor_series(1,3)
...e-82 + (...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3)


Check that trac ticket #25402 is fixed:

sage: L = EllipticCurve("24a1").modular_form().lseries()
sage: L.taylor_series(-1, 3)
0.000000000000000 - 0.702565506265199*z + 0.638929001045535*z^2 + O(z^3)


Check that trac ticket #25965 is fixed:

sage: L2 = EllipticCurve("37a1").modular_form().lseries(); L2
L-series associated to the cusp form q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6)
sage: L2.taylor_series(0,4)
0.000000000000000 - 0.357620466127498*z + 0.273373112603865*z^2 + 0.303362857047671*z^3 + O(z^4)
sage: L2.taylor_series(0,1)
O(z^1)
sage: L2(0)
0.000000000000000