LMFDB - L-function 2-5328-37.36-c1-0-29

Dirichlet series

L(s) = 1

− 0.540i·5^-s + 1.23·7^-s + 2.47·11^-s + 4.57i·13^-s − 3.36i·17^-s + 1.74i·19^-s + 8.61i·23^-s + 4.70·25^-s − 7.94i·29^-s + 6.32i·31^-s − 0.667i·35^-s + (−2.23 + 5.65i)37^-s − 2·41^-s + 2.82i·43^-s + 4·47^-s + ⋯

L(s) = 1

− 0.241i·5^-s + 0.467·7^-s + 0.745·11^-s + 1.26i·13^-s − 0.817i·17^-s + 0.401i·19^-s + 1.79i·23^-s + 0.941·25^-s − 1.47i·29^-s + 1.13i·31^-s − 0.112i·35^-s + (−0.367 + 0.929i)37^-s − 0.312·41^-s + 0.431i·43^-s + 0.583·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5328$ = $2^{4} \cdot 3^{2} \cdot 37$
Sign:	$0.367 - 0.929i$
Analytic conductor:	$42.5442$
Root analytic conductor:	$6.52259$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5328} (2737, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5328,\ (\ :1/2),\ 0.367 - 0.929i)$

Particular Values

$L(1)$	$\approx$	$1.923168196$
$L(\frac12)$	$\approx$	$1.923168196$
$L(\frac{3}{2})$		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 $
	37	$ 1 + (2.23 - 5.65i)T $
good	5	$ 1 + 0.540iT - 5T^{2} $
	7	$ 1 - 1.23T + 7T^{2} $
	11	$ 1 - 2.47T + 11T^{2} $
	13	$ 1 - 4.57iT - 13T^{2} $
	17	$ 1 + 3.36iT - 17T^{2} $
	19	$ 1 - 1.74iT - 19T^{2} $
	23	$ 1 - 8.61iT - 23T^{2} $
	29	$ 1 + 7.94iT - 29T^{2} $
	31	$ 1 - 6.32iT - 31T^{2} $
	41	$ 1 + 2T + 41T^{2} $
	43	$ 1 - 2.82iT - 43T^{2} $
	47	$ 1 - 4T + 47T^{2} $
	53	$ 1 + 10.9T + 53T^{2} $
	59	$ 1 - 2.28iT - 59T^{2} $
	61	$ 1 + 5.65iT - 61T^{2} $
	67	$ 1 - 7.70T + 67T^{2} $
	71	$ 1 + 8.94T + 71T^{2} $
	73	$ 1 - 3.23T + 73T^{2} $
	79	$ 1 - 9.56iT - 79T^{2} $
	83	$ 1 - 1.52T + 83T^{2} $
	89	$ 1 - 8.61iT - 89T^{2} $
	97	$ 1 + 13.7iT - 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

−8.299677748738159581663857834509, −7.63807904712209696937384015517, −6.82891293123246841948919931292, −6.32152797520581935951835861019, −5.27327086332295005008639872001, −4.71106040146817367187824793357, −3.91789653886722751269097007251, −3.07458377140746897142825149665, −1.86588739498803326529987840661, −1.18682585185619962889424152924, 0.54150987573453156244915303246, 1.67079387137310440153767295792, 2.71461668321230136717715315945, 3.49405439811996168443988026298, 4.40619621260101854226819849963, 5.09973056035468004875529107441, 5.95556764502630114290514498569, 6.61654960079336109074139066420, 7.31785453109684594322747866656, 8.141531567167490593853840646099

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.923168196\)
\(L(\frac12)\)	\(\approx\)	\(1.923168196\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more