LMFDB - L-function 2-78e2-13.12-c1-0-26

Dirichlet series

L(s) = 1

+ 2.64i·5^-s + 2i·7^-s − 5.29i·11^-s − 7.93·17^-s − 6i·19^-s + 5.29·23^-s − 2.00·25^-s + 2.64·29^-s + 4i·31^-s − 5.29·35^-s + 3i·37^-s + 7.93i·41^-s + 2·43^-s − 5.29i·47^-s + 3·49^-s + ⋯

L(s) = 1

+ 1.18i·5^-s + 0.755i·7^-s − 1.59i·11^-s − 1.92·17^-s − 1.37i·19^-s + 1.10·23^-s − 0.400·25^-s + 0.491·29^-s + 0.718i·31^-s − 0.894·35^-s + 0.493i·37^-s + 1.23i·41^-s + 0.304·43^-s − 0.771i·47^-s + 0.428·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6084$ = $2^{2} \cdot 3^{2} \cdot 13^{2}$
Sign:	$0.832 - 0.554i$
Analytic conductor:	$48.5809$
Root analytic conductor:	$6.97000$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6084} (4393, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6084,\ (\ :1/2),\ 0.832 - 0.554i)$

Particular Values

$L(1)$	$\approx$	$1.760390610$
$L(\frac12)$	$\approx$	$1.760390610$
$L(\frac{3}{2})$		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 $
	13	$ 1 $
good	5	$ 1 - 2.64iT - 5T^{2} $
	7	$ 1 - 2iT - 7T^{2} $
	11	$ 1 + 5.29iT - 11T^{2} $
	17	$ 1 + 7.93T + 17T^{2} $
	19	$ 1 + 6iT - 19T^{2} $
	23	$ 1 - 5.29T + 23T^{2} $
	29	$ 1 - 2.64T + 29T^{2} $
	31	$ 1 - 4iT - 31T^{2} $
	37	$ 1 - 3iT - 37T^{2} $
	41	$ 1 - 7.93iT - 41T^{2} $
	43	$ 1 - 2T + 43T^{2} $
	47	$ 1 + 5.29iT - 47T^{2} $
	53	$ 1 - 7.93T + 53T^{2} $
	59	$ 1 + 10.5iT - 59T^{2} $
	61	$ 1 - 13T + 61T^{2} $
	67	$ 1 + 2iT - 67T^{2} $
	71	$ 1 - 5.29iT - 71T^{2} $
	73	$ 1 - 7iT - 73T^{2} $
	79	$ 1 + 4T + 79T^{2} $
	83	$ 1 + 15.8iT - 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 2iT - 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

−8.419188823677209357125077658079, −7.14117125961160430318399680623, −6.72194648808355840947704634348, −6.20527302271423644290530842961, −5.30743652657570578768211416349, −4.58146860233411240892781943507, −3.44553340141929751749511229439, −2.82081095737370415561386707199, −2.28995213211773416978028764483, −0.70145465678015439861907802134, 0.68224575516304149380698444799, 1.70087814950988463366599004605, 2.47414777257690956838102414481, 4.00881949301678191272096330797, 4.27374362121762830318721838309, 4.98082451996679712795097714269, 5.75745398886677038213694393374, 6.81738181849805058682072567329, 7.21582222640926262336251573311, 8.001422117862306524346951103087

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.760390610\)
\(L(\frac12)\)	\(\approx\)	\(1.760390610\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more