LMFDB - L-function 2-864-24.11-c3-0-22

Dirichlet series

L(s) = 1

− 9.32·5^-s + 5.38i·7^-s − 53.5i·11^-s + 57.0i·13^-s + 38.0i·17^-s − 75.4·19^-s + 98.3·23^-s − 38.0·25^-s + 11.9·29^-s + 94.3i·31^-s − 50.2i·35^-s − 423. i·37^-s + 158. i·41^-s − 403.·43^-s + 119.·47^-s + ⋯

L(s) = 1

− 0.833·5^-s + 0.290i·7^-s − 1.46i·11^-s + 1.21i·13^-s + 0.542i·17^-s − 0.910·19^-s + 0.891·23^-s − 0.304·25^-s + 0.0766·29^-s + 0.546i·31^-s − 0.242i·35^-s − 1.88i·37^-s + 0.605i·41^-s − 1.42·43^-s + 0.371·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.925 + 0.379i$
Analytic conductor:	$50.9776$
Root analytic conductor:	$7.13986$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :3/2),\ 0.925 + 0.379i)$

Particular Values

$L(2)$	$\approx$	$1.374009525$
$L(\frac12)$	$\approx$	$1.374009525$
$L(\frac{5}{2})$		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 + 9.32T + 125T^{2} $
	7	$ 1 - 5.38iT - 343T^{2} $
	11	$ 1 + 53.5iT - 1.33e3T^{2} $
	13	$ 1 - 57.0iT - 2.19e3T^{2} $
	17	$ 1 - 38.0iT - 4.91e3T^{2} $
	19	$ 1 + 75.4T + 6.85e3T^{2} $
	23	$ 1 - 98.3T + 1.21e4T^{2} $
	29	$ 1 - 11.9T + 2.43e4T^{2} $
	31	$ 1 - 94.3iT - 2.97e4T^{2} $
	37	$ 1 + 423. iT - 5.06e4T^{2} $
	41	$ 1 - 158. iT - 6.89e4T^{2} $
	43	$ 1 + 403.T + 7.95e4T^{2} $
	47	$ 1 - 119.T + 1.03e5T^{2} $
	53	$ 1 - 425.T + 1.48e5T^{2} $
	59	$ 1 + 616. iT - 2.05e5T^{2} $
	61	$ 1 + 58.7iT - 2.26e5T^{2} $
	67	$ 1 - 346.T + 3.00e5T^{2} $
	71	$ 1 - 844.T + 3.57e5T^{2} $
	73	$ 1 - 918.T + 3.89e5T^{2} $
	79	$ 1 - 490. iT - 4.93e5T^{2} $
	83	$ 1 - 1.22e3iT - 5.71e5T^{2} $
	89	$ 1 + 549. iT - 7.04e5T^{2} $
	97	$ 1 - 301.T + 9.12e5T^{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

−9.599939064616610127215611141881, −8.658249292191114265878690106895, −8.289980378046820749625798851167, −7.10665488074160306072377114990, −6.32628348275006978678353712073, −5.33259900172249258652102653930, −4.14098147020088766454233173503, −3.44314309005277113063998621365, −2.08458292618771800228556440470, −0.55352266195023711461926413378, 0.72846235869760570909722249181, 2.27784728484941708555833129161, 3.47786915243117129684913508386, 4.45954551441251823771501010814, 5.23062539759552323676157386679, 6.57139314957153769113390089620, 7.37027718012394230855019199357, 7.991001478972269312619334100873, 8.925557177079031656098525914738, 10.02263722693287381656483080646

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.374009525\)
\(L(\frac12)\)	\(\approx\)	\(1.374009525\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more