LMFDB - L-function 2-4032-24.11-c1-0-10

Dirichlet series

L(s) = 1

− 0.646·5^-s + i·7^-s + 3.09i·11^-s − 0.913i·13^-s + 6.77i·17^-s + 5.29·19^-s + 2.53·23^-s − 4.58·25^-s + 9.93·29^-s − 9.16i·31^-s − 0.646i·35^-s + 7.84i·37^-s − 9.01i·41^-s − 12.2·43^-s − 5.65·47^-s + ⋯

L(s) = 1

− 0.288·5^-s + 0.377i·7^-s + 0.933i·11^-s − 0.253i·13^-s + 1.64i·17^-s + 1.21·19^-s + 0.528·23^-s − 0.916·25^-s + 1.84·29^-s − 1.64i·31^-s − 0.109i·35^-s + 1.28i·37^-s − 1.40i·41^-s − 1.86·43^-s − 0.825·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.346 - 0.938i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.346 - 0.938i)$

Particular Values

$L(1)$	$\approx$	$1.347484857$
$L(\frac12)$	$\approx$	$1.347484857$
$L(\frac{3}{2})$		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 $
	7	$ 1 - iT $
good	5	$ 1 + 0.646T + 5T^{2} $
	11	$ 1 - 3.09iT - 11T^{2} $
	13	$ 1 + 0.913iT - 13T^{2} $
	17	$ 1 - 6.77iT - 17T^{2} $
	19	$ 1 - 5.29T + 19T^{2} $
	23	$ 1 - 2.53T + 23T^{2} $
	29	$ 1 - 9.93T + 29T^{2} $
	31	$ 1 + 9.16iT - 31T^{2} $
	37	$ 1 - 7.84iT - 37T^{2} $
	41	$ 1 + 9.01iT - 41T^{2} $
	43	$ 1 + 12.2T + 43T^{2} $
	47	$ 1 + 5.65T + 47T^{2} $
	53	$ 1 - 1.15T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 - 6.92iT - 61T^{2} $
	67	$ 1 + 7.84T + 67T^{2} $
	71	$ 1 + 5.36T + 71T^{2} $
	73	$ 1 + 0.417T + 73T^{2} $
	79	$ 1 - 10.7iT - 79T^{2} $
	83	$ 1 - 15.9iT - 83T^{2} $
	89	$ 1 - 9.01iT - 89T^{2} $
	97	$ 1 + 11.5T + 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

−8.420738540258208089560710645429, −8.094999903553939886077928132422, −7.22114126096701288704834347644, −6.48153144695275392054250109011, −5.70756001808077761159261549558, −4.90768364211216023826481875287, −4.10703123582461759917581470580, −3.26891414713724701425839374804, −2.26294584857771602178936975017, −1.25213980044446213158122507865, 0.41740603356420236361077799977, 1.46922200587654134678064935254, 3.05233875526128355099073309012, 3.25409970488395035486957139530, 4.61137398143543779293046210165, 5.05645708846153740756109955906, 6.04939183647311939813269400566, 6.87480990532988639549663425582, 7.42365506656986178601500942116, 8.232153323270213602399824451408

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.347484857\)
\(L(\frac12)\)	\(\approx\)	\(1.347484857\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more