LMFDB - L-function 2-2880-8.5-c1-0-29

Dirichlet series

L(s) = 1

− i·5^-s − 3.46·7^-s + 3.46i·11^-s + 6.92·17^-s − 2i·19^-s − 6·23^-s − 25^-s − 6i·29^-s + 6.92·31^-s + 3.46i·35^-s + 6.92i·37^-s − 6.92·41^-s − 4i·43^-s − 6·47^-s + 4.99·49^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 1.30·7^-s + 1.04i·11^-s + 1.68·17^-s − 0.458i·19^-s − 1.25·23^-s − 0.200·25^-s − 1.11i·29^-s + 1.24·31^-s + 0.585i·35^-s + 1.13i·37^-s − 1.08·41^-s − 0.609i·43^-s − 0.875·47^-s + 0.714·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2880$ = $2^{6} \cdot 3^{2} \cdot 5$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$22.9969$
Root analytic conductor:	$4.79550$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2880} (1441, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2880,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.6184560736$
$L(\frac12)$	$\approx$	$0.6184560736$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
good	7	$ 1 + 3.46T + 7T^{2} $
	11	$ 1 - 3.46iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 6.92T + 17T^{2} $
	19	$ 1 + 2iT - 19T^{2} $
	23	$ 1 + 6T + 23T^{2} $
	29	$ 1 + 6iT - 29T^{2} $
	31	$ 1 - 6.92T + 31T^{2} $
	37	$ 1 - 6.92iT - 37T^{2} $
	41	$ 1 + 6.92T + 41T^{2} $
	43	$ 1 + 4iT - 43T^{2} $
	47	$ 1 + 6T + 47T^{2} $
	53	$ 1 + 6iT - 53T^{2} $
	59	$ 1 + 10.3iT - 59T^{2} $
	61	$ 1 + 13.8iT - 61T^{2} $
	67	$ 1 - 8iT - 67T^{2} $
	71	$ 1 + 12T + 71T^{2} $
	73	$ 1 - 2T + 73T^{2} $
	79	$ 1 + 6.92T + 79T^{2} $
	83	$ 1 + 6.92iT - 83T^{2} $
	89	$ 1 + 6.92T + 89T^{2} $
	97	$ 1 + 14T + 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

−8.360856961651924341082999644455, −7.85413316186768129327969343288, −6.85726655174840718677569041039, −6.29689197454085270527862757793, −5.41631351465746174666060196957, −4.56407856206862881195630493535, −3.64924711182703012214796736622, −2.83180644050823276187235041427, −1.63564442881427955586536715276, −0.20632366111293867949882281991, 1.24674177743422189053078997279, 2.82577109003552860306307379610, 3.30683945468763352187193531920, 4.10786555906113252774288593088, 5.52947927953441364970317853856, 5.97451802258817382971084136297, 6.69292721510648225972978796715, 7.55713612353509473015518615495, 8.268759180509311817136889814927, 9.080012061276202808687534215059

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6184560736\)
\(L(\frac12)\)	\(\approx\)	\(0.6184560736\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more