LMFDB - L-function 2-3312-92.91-c1-0-2

Dirichlet series

L(s) = 1

− 2i·5^-s − 3.38·7^-s + 4.73·11^-s − 3.46·13^-s + 0.732i·17^-s − 2.47·19^-s + (1.26 − 4.62i)23^-s + 25^-s − 4.29·29^-s + 6i·31^-s + 6.77i·35^-s − 10.1i·37^-s − 11.7·41^-s + 2.47·43^-s + 4.29i·47^-s + ⋯

L(s) = 1

− 0.894i·5^-s − 1.27·7^-s + 1.42·11^-s − 0.960·13^-s + 0.177i·17^-s − 0.568·19^-s + (0.264 − 0.964i)23^-s + 0.200·25^-s − 0.797·29^-s + 1.07i·31^-s + 1.14i·35^-s − 1.66i·37^-s − 1.83·41^-s + 0.377·43^-s + 0.626i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3312$ = $2^{4} \cdot 3^{2} \cdot 23$
Sign:	$-0.703 - 0.711i$
Analytic conductor:	$26.4464$
Root analytic conductor:	$5.14261$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3312} (2575, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3312,\ (\ :1/2),\ -0.703 - 0.711i)$

Particular Values

$L(1)$	$\approx$	$0.1201776747$
$L(\frac12)$	$\approx$	$0.1201776747$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	23	$ 1 + (-1.26 + 4.62i)T $
good	5	$ 1 + 2iT - 5T^{2} $
	7	$ 1 + 3.38T + 7T^{2} $
	11	$ 1 - 4.73T + 11T^{2} $
	13	$ 1 + 3.46T + 13T^{2} $
	17	$ 1 - 0.732iT - 17T^{2} $
	19	$ 1 + 2.47T + 19T^{2} $
	29	$ 1 + 4.29T + 29T^{2} $
	31	$ 1 - 6iT - 31T^{2} $
	37	$ 1 + 10.1iT - 37T^{2} $
	41	$ 1 + 11.7T + 41T^{2} $
	43	$ 1 - 2.47T + 43T^{2} $
	47	$ 1 - 4.29iT - 47T^{2} $
	53	$ 1 + 0.535iT - 53T^{2} $
	59	$ 1 - 11.7iT - 59T^{2} $
	61	$ 1 + 1.57iT - 61T^{2} $
	67	$ 1 - 9.25T + 67T^{2} $
	71	$ 1 + 1.81iT - 71T^{2} $
	73	$ 1 + 12.3T + 73T^{2} $
	79	$ 1 + 8.34T + 79T^{2} $
	83	$ 1 + 2.19T + 83T^{2} $
	89	$ 1 - 13.1iT - 89T^{2} $
	97	$ 1 - 11.7iT - 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

−9.010837326903171824630265104935, −8.417643769602107575807232180820, −7.19917686937214755616962014155, −6.73248242265263494901258620986, −5.97358603759247762177503353294, −5.05829081629213714108551695754, −4.23253238800825984126490609887, −3.50910872468957097861256808618, −2.42737193613006108399713276384, −1.20679287847891330355898626414, 0.03775012616564935048990341936, 1.71109982908279803839481416829, 2.88740927481633134062313685298, 3.45969510861603285934228492404, 4.31117977029481818716413775539, 5.43710638948148881830432109316, 6.37615527564253169061584073285, 6.79380244015827439907044402223, 7.32021196907294943586243838618, 8.415048900033840515098484146269

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1201776747\)
\(L(\frac12)\)	\(\approx\)	\(0.1201776747\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more