LMFDB - L-function 2-1152-8.5-c3-0-7

Dirichlet series

L(s) = 1

− 17.4i·5^-s + 2.99·7^-s − 10.6i·11^-s + 43.3i·13^-s + 37.8·17^-s + 79.8i·19^-s − 191.·23^-s − 178.·25^-s + 138. i·29^-s − 212.·31^-s − 52.1i·35^-s − 270. i·37^-s + 441.·41^-s − 64.1i·43^-s − 436.·47^-s + ⋯

L(s) = 1

− 1.55i·5^-s + 0.161·7^-s − 0.291i·11^-s + 0.924i·13^-s + 0.540·17^-s + 0.964i·19^-s − 1.73·23^-s − 1.43·25^-s + 0.889i·29^-s − 1.22·31^-s − 0.251i·35^-s − 1.20i·37^-s + 1.68·41^-s − 0.227i·43^-s − 1.35·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$-i$
Analytic conductor:	$67.9702$
Root analytic conductor:	$8.24440$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :3/2),\ -i)$

Particular Values

$L(2)$	$\approx$	$0.7693680781$
$L(\frac12)$	$\approx$	$0.7693680781$
$L(\frac{5}{2})$		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 + 17.4iT - 125T^{2} $
	7	$ 1 - 2.99T + 343T^{2} $
	11	$ 1 + 10.6iT - 1.33e3T^{2} $
	13	$ 1 - 43.3iT - 2.19e3T^{2} $
	17	$ 1 - 37.8T + 4.91e3T^{2} $
	19	$ 1 - 79.8iT - 6.85e3T^{2} $
	23	$ 1 + 191.T + 1.21e4T^{2} $
	29	$ 1 - 138. iT - 2.43e4T^{2} $
	31	$ 1 + 212.T + 2.97e4T^{2} $
	37	$ 1 + 270. iT - 5.06e4T^{2} $
	41	$ 1 - 441.T + 6.89e4T^{2} $
	43	$ 1 + 64.1iT - 7.95e4T^{2} $
	47	$ 1 + 436.T + 1.03e5T^{2} $
	53	$ 1 + 278. iT - 1.48e5T^{2} $
	59	$ 1 - 830. iT - 2.05e5T^{2} $
	61	$ 1 - 724. iT - 2.26e5T^{2} $
	67	$ 1 - 859. iT - 3.00e5T^{2} $
	71	$ 1 - 681.T + 3.57e5T^{2} $
	73	$ 1 + 785.T + 3.89e5T^{2} $
	79	$ 1 - 1.01e3T + 4.93e5T^{2} $
	83	$ 1 + 467. iT - 5.71e5T^{2} $
	89	$ 1 + 510.T + 7.04e5T^{2} $
	97	$ 1 + 234.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

−9.460108207073213029459648292145, −8.824665832507377112370772222208, −8.118771112546601554895089009731, −7.33380364390231259583236040319, −5.98955793832954602191244295529, −5.44082219097473619651159886351, −4.38382548397458244091819551848, −3.73383295787935081753443259970, −2.01149314010461355207566147560, −1.16329946430623431297983349548, 0.18704147047753559682033196515, 1.95623543902953692579732440371, 2.92298887870634147017861010240, 3.71310039081462585177292609242, 4.94544536973608910052291060768, 6.06038209565909892675570533451, 6.62922633347270562661977082128, 7.71442614369701921979922180986, 8.016483398229960459113594631876, 9.545390623722643941518302681208

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.7693680781\)
\(L(\frac12)\)	\(\approx\)	\(0.7693680781\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more