L-functions

(PARI-GP version 2.11.0)

Characters

A character on the abelian group G = ∑_{j≤k}(Z/d_{j}Z)·g_{j}, e.g. from znstar(q,1) ↔ (Z/qZ)^{∗} or bnrinit ↔ Cl𝔣(K), is coded by χ = [c_{1}, …, c_{k}] such that χ(g_{j}) = e(c_{j}/d_{j}). Our L-functions consider the attached primitive character. Dirichlet characters χ_{q}(m, ·) in Conrey labelling system are alternatively concisely coded by Mod(m,q). Finally, a quadratic character (D/·) can also be coded by the integer D.

L-function Constructors

An Ldata is a GP structure describing the functional equation for L(s) = ∑_{n≥1} a_{n}n^{−s}.

• Dirichlet coefficients given by closure a : N ↦ [a_{1}, …, a_{N}].

• Dirichlet coefficients a^{∗}(n) for dual L-function L^{∗}.

• Euler factor A = [a_{1}, …, a_{d}] for γ_{A}(s) = ∏_{i} ΓR(s + a_{i}),

• classical weight k (values at s and k − s are related),

• conductor N, Λ(s) = N^{s/2}γ_{A}(s),

• root number ε; Λ(a, k − s) = εΛ(a^{∗}, s).

• polar part: list of [β, P_{β}(x)].

An Linit is a GP structure containing an Ldata L and an evaluation domain fixing a maximal order of derivation m and bit accuracy (realbitprecision), together with complex ranges

• for L-function: R = [c, w, h] (coding |ℜz − c| ≤ w, |ℑz| ≤ h); or R = [w, h] (for c = k/2); or R = [h] (for c = k/2, w = 0).

• for θ-function: T = [ρ, α] (for |t| ≥ ρ, | arg t| ≤ α); or T = ρ (for α = 0). Ldata constructors Riemann ζ lfuncreate(1) Dirichlet for quadratic char. (D/·) lfuncreate(D) Dirichlet series L(χ_{q}(m, ·), s) lfuncreate(Mod(m,q)) Dedekind ζ_{K}, K = Q[x]/(T) lfuncreate(bnf ), lfuncreate(T) Hecke for χ mod 𝔣 lfuncreate([bnr, χ]) Artin L-function lfunartin(nf , gal, M, n) Lattice Θ-function lfunqf(Q) From eigenform F lfunmf(F) Quotients of Dedekind η: ∏_{i} η(m_{i,1} · τ)^{m_{i,2}} lfunetaquo(M) L(E, s), E elliptic curve E = ellinit(…) L(Sym^{m}E, s), E elliptic curve lfunsympow(E, m) genus 2 curve, y^{2} + xQ = P lfungenus2([P, Q])

L_{1} · L_{2} lfunmul(L_{1}, L_{2}) L_{1}/L_{2} lfundiv(L_{1}, L_{2}) twist by Dirichlet character lfuntwist(L, χ)

low-level constructor lfuncreate([a, a^{∗}, A, k, N, eps, poles]) check functional equation (at t) lfuncheckfeq(L, {t}) Linit constructors initialize for L lfuninit(L, R, {m = 0}) initialize for θ lfunthetainit(L, {T = 1}, {m = 0}) cost of lfuninit lfuncost(L, R, {m = 0})

cost of lfunthetainit            lfunthetacost(L, T, {m = 0})

Dedekind ζ_{L}, L abelian over a subfield lfunabelianrelinit

L-functions

L is either an Ldata or an Linit (more efficient for many values).

Evaluate

L^{(k)}(s) lfun(L, s, {k = 0}) Λ^{(k)}(s) lfunlambda(L, s, {k = 0}) θ^{(k)}(t) lfuntheta(L, t, {k = 0}) generalized Hardy Z-function at t lfunhardy(L, t) Zeros order of zero at s = k/2 lfunorderzero(L, {m = −1}) zeros s = k/2 + it, 0 ≤ t ≤ T lfunzeros(L, T, {h}) Dirichlet series and functional equation [a_{n}: 1 ≤ n ≤ N] lfunan(L, N) conductor N of L lfunconductor(L) root number and residues lfunrootres(L) G-functions Attached to inverse Mellin transform for γ_{A}(s), A = [a_{1}, …, a_{d}]. initialize for G attached to A gammamellininvinit(A) G^{(k)}(t) gammamellininv(G, t, {k = 0}) asymp. expansion of G^{(k)}(t) gammamellininvasymp(A, n, {k = 0})

Based on an earlier version by Joseph H. Silverman July 2018 v2.35. Copyright \begin{matrix}c \\ ◯ \end{matrix} 2018 K. Belabas Permission is granted to make and distribute copies of this card provided the copyright and this permission notice are preserved on all copies. Send comments and corrections to ⟨Karim.Belabas@math.u-bordeaux.fr⟩