LMFDB - L-function 2-84e2-28.27-c1-0-56

Dirichlet series

L(s) = 1

− 3.87i·5^-s + 3.87i·11^-s + 3.46i·13^-s − 4·19^-s + 7.74i·23^-s − 10.0·25^-s − 6.70·29^-s + 31^-s + 4·37^-s − 7.74i·41^-s − 6.92i·43^-s + 13.4·47^-s + 6.70·53^-s + 15.0·55^-s + 6.70·59^-s + ⋯

L(s) = 1

− 1.73i·5^-s + 1.16i·11^-s + 0.960i·13^-s − 0.917·19^-s + 1.61i·23^-s − 2.00·25^-s − 1.24·29^-s + 0.179·31^-s + 0.657·37^-s − 1.20i·41^-s − 1.05i·43^-s + 1.95·47^-s + 0.921·53^-s + 2.02·55^-s + 0.873·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.188 + 0.981i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ 0.188 + 0.981i)$

Particular Values

$L(1)$	$\approx$	$1.561516164$
$L(\frac12)$	$\approx$	$1.561516164$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.87iT - 5T^{2} $
	11	$ 1 - 3.87iT - 11T^{2} $
	13	$ 1 - 3.46iT - 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 4T + 19T^{2} $
	23	$ 1 - 7.74iT - 23T^{2} $
	29	$ 1 + 6.70T + 29T^{2} $
	31	$ 1 - T + 31T^{2} $
	37	$ 1 - 4T + 37T^{2} $
	41	$ 1 + 7.74iT - 41T^{2} $
	43	$ 1 + 6.92iT - 43T^{2} $
	47	$ 1 - 13.4T + 47T^{2} $
	53	$ 1 - 6.70T + 53T^{2} $
	59	$ 1 - 6.70T + 59T^{2} $
	61	$ 1 + 10.3iT - 61T^{2} $
	67	$ 1 + 6.92iT - 67T^{2} $
	71	$ 1 + 7.74iT - 71T^{2} $
	73	$ 1 + 6.92iT - 73T^{2} $
	79	$ 1 + 12.1iT - 79T^{2} $
	83	$ 1 - 6.70T + 83T^{2} $
	89	$ 1 + 7.74iT - 89T^{2} $
	97	$ 1 + 5.19iT - 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

−7.63555921968537644120833682982, −7.36524919162892742280821527228, −6.30541139588341917607408480130, −5.49132887716754456403200517522, −4.96393884382612474219096136768, −4.18798816776879747443436082645, −3.77460678352694651819785755503, −2.05792504553396729337691159715, −1.75849633751597617554734064981, −0.47793205240919484970007725540, 0.829012333654465463349074760160, 2.51682842873814920428094647145, 2.68610099673332523804600045851, 3.66153651414517251206560734451, 4.29437340775562128654432464710, 5.63045393954638172188494618209, 5.98608650284893970265327598413, 6.72184880071494449046264645789, 7.27085684382986207323463180716, 8.143358259088232815652560548124

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.561516164\)
\(L(\frac12)\)	\(\approx\)	\(1.561516164\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more