LMFDB - L-function 2-3042-13.12-c1-0-51

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s − 0.267i·5^-s − 0.732i·7^-s + i·8^-s − 0.267·10^-s + 4.73i·11^-s − 0.732·14^-s + 16^-s − 2.26·17^-s + 1.26i·19^-s + 0.267i·20^-s + 4.73·22^-s − 6.19·23^-s + 4.92·25^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.119i·5^-s − 0.276i·7^-s + 0.353i·8^-s − 0.0847·10^-s + 1.42i·11^-s − 0.195·14^-s + 0.250·16^-s − 0.550·17^-s + 0.290i·19^-s + 0.0599i·20^-s + 1.00·22^-s − 1.29·23^-s + 0.985·25^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3042$ = $2 \cdot 3^{2} \cdot 13^{2}$
Sign:	$-0.960 + 0.277i$
Analytic conductor:	$24.2904$
Root analytic conductor:	$4.92853$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3042} (1351, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3042,\ (\ :1/2),\ -0.960 + 0.277i)$

Particular Values

$L(1)$	$\approx$	$0.8026172892$
$L(\frac12)$	$\approx$	$0.8026172892$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 + iT $
	3	$ 1 $
	13	$ 1 $
good	5	$ 1 + 0.267iT - 5T^{2} $
	7	$ 1 + 0.732iT - 7T^{2} $
	11	$ 1 - 4.73iT - 11T^{2} $
	17	$ 1 + 2.26T + 17T^{2} $
	19	$ 1 - 1.26iT - 19T^{2} $
	23	$ 1 + 6.19T + 23T^{2} $
	29	$ 1 + 2.46T + 29T^{2} $
	31	$ 1 + 5.46iT - 31T^{2} $
	37	$ 1 + 10.4iT - 37T^{2} $
	41	$ 1 + 11.3iT - 41T^{2} $
	43	$ 1 + 7.66T + 43T^{2} $
	47	$ 1 + 8.19iT - 47T^{2} $
	53	$ 1 + 0.464T + 53T^{2} $
	59	$ 1 - 8iT - 59T^{2} $
	61	$ 1 - 1.19T + 61T^{2} $
	67	$ 1 + 11.1iT - 67T^{2} $
	71	$ 1 + 1.26iT - 71T^{2} $
	73	$ 1 - 9.73iT - 73T^{2} $
	79	$ 1 + 9.46T + 79T^{2} $
	83	$ 1 + 10.1iT - 83T^{2} $
	89	$ 1 + 2.53iT - 89T^{2} $
	97	$ 1 + 6iT - 97T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.594481150303802868182784752399, −7.53862578667208030779271220572, −7.09094581939577671757210439272, −5.97920201980613615158359463612, −5.13249255116510348281264422059, −4.24551169775977822562442416109, −3.75127391137503807564301600127, −2.36840272403895561543210616928, −1.79636515511229659383477890065, −0.26085182800569717763175867203, 1.22908884002359873655489298257, 2.73994971371760346649848996072, 3.51452518265488197486432134784, 4.58682070005904547483214069612, 5.30487877025916276875374820309, 6.26083451395660273360724818217, 6.54685230359560823149645254826, 7.62209353850871041828246016213, 8.418488972189044625916330325170, 8.716727167578236702309870022480

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8026172892\)
\(L(\frac12)\)	\(\approx\)	\(0.8026172892\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more