LMFDB - L-function 2-864-24.11-c1-0-5

Dirichlet series

L(s) = 1

+ 3.07·5^-s + 3.99i·7^-s + 1.89i·11^-s − 1.06i·13^-s + 7.08i·17^-s − 3.73·19^-s + 0.824·23^-s + 4.46·25^-s − 4.50·29^-s − 5.84i·31^-s + 12.2i·35^-s − 6.91i·37^-s + 3.79i·41^-s − 2·43^-s + 6.97·47^-s + ⋯

L(s) = 1

+ 1.37·5^-s + 1.50i·7^-s + 0.572i·11^-s − 0.296i·13^-s + 1.71i·17^-s − 0.856·19^-s + 0.171·23^-s + 0.892·25^-s − 0.836·29^-s − 1.04i·31^-s + 2.07i·35^-s − 1.13i·37^-s + 0.593i·41^-s − 0.304·43^-s + 1.01·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.398 - 0.917i$
Analytic conductor:	$6.89907$
Root analytic conductor:	$2.62660$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :1/2),\ 0.398 - 0.917i)$

Particular Values

$L(1)$	$\approx$	$1.55838 + 1.02249i$
$L(\frac12)$	$\approx$	$1.55838 + 1.02249i$
$L(\frac{3}{2})$		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 $
good	5	$ 1 - 3.07T + 5T^{2} $
	7	$ 1 - 3.99iT - 7T^{2} $
	11	$ 1 - 1.89iT - 11T^{2} $
	13	$ 1 + 1.06iT - 13T^{2} $
	17	$ 1 - 7.08iT - 17T^{2} $
	19	$ 1 + 3.73T + 19T^{2} $
	23	$ 1 - 0.824T + 23T^{2} $
	29	$ 1 + 4.50T + 29T^{2} $
	31	$ 1 + 5.84iT - 31T^{2} $
	37	$ 1 + 6.91iT - 37T^{2} $
	41	$ 1 - 3.79iT - 41T^{2} $
	43	$ 1 + 2T + 43T^{2} $
	47	$ 1 - 6.97T + 47T^{2} $
	53	$ 1 - 10.6T + 53T^{2} $
	59	$ 1 - 1.89iT - 59T^{2} $
	61	$ 1 - 1.06iT - 61T^{2} $
	67	$ 1 - 9.19T + 67T^{2} $
	71	$ 1 - 10.6T + 71T^{2} $
	73	$ 1 - 5.92T + 73T^{2} $
	79	$ 1 + 6.12iT - 79T^{2} $
	83	$ 1 + 10.3iT - 83T^{2} $
	89	$ 1 - 13.6iT - 89T^{2} $
	97	$ 1 + 11.3T + 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

−10.22204018306662256132081922666, −9.417164335487847629457057343418, −8.827896844658364710462824428455, −7.952813415771305086722059877238, −6.58482650264884970151224363224, −5.84141121148869139293340690235, −5.39553580231983850475358844037, −3.99570250662774697839360039015, −2.41007025508879097966658361544, −1.89838186534237206971519292689, 0.920128894009382081361876497853, 2.28706795829768838864160004102, 3.56211286921638146689124051728, 4.72925713545386785940653909487, 5.60365012255747419478724137631, 6.71311165612538271639004740583, 7.17307557454836959043598891805, 8.421758359011326255289557253532, 9.360812968945081698904393653022, 9.995779295847502170615878002121

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.55838 + 1.02249i\)
\(L(\frac12)\)	\(\approx\)	\(1.55838 + 1.02249i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more