LMFDB - L-function 2-3168-88.43-c1-0-38

Dirichlet series

L(s) = 1

− 2i·5^-s + 3.02·7^-s + (2.33 + 2.35i)11^-s − 1.32·13^-s + 1.69i·17^-s − 7.04i·19^-s − 3.12i·23^-s + 25^-s + 1.54·29^-s − 8.30i·31^-s − 6.04i·35^-s − 4.66i·37^-s + 7.73i·41^-s + 3.95i·43^-s + 6i·47^-s + ⋯

L(s) = 1

− 0.894i·5^-s + 1.14·7^-s + (0.703 + 0.711i)11^-s − 0.367·13^-s + 0.411i·17^-s − 1.61i·19^-s − 0.651i·23^-s + 0.200·25^-s + 0.286·29^-s − 1.49i·31^-s − 1.02i·35^-s − 0.766i·37^-s + 1.20i·41^-s + 0.603i·43^-s + 0.875i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3168$ = $2^{5} \cdot 3^{2} \cdot 11$
Sign:	$0.517 + 0.855i$
Analytic conductor:	$25.2966$
Root analytic conductor:	$5.02957$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3168} (2287, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3168,\ (\ :1/2),\ 0.517 + 0.855i)$

Particular Values

$L(1)$	$\approx$	$2.190455927$
$L(\frac12)$	$\approx$	$2.190455927$
$L(\frac{3}{2})$		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 $
	11	$ 1 + (-2.33 - 2.35i)T $
good	5	$ 1 + 2iT - 5T^{2} $
	7	$ 1 - 3.02T + 7T^{2} $
	13	$ 1 + 1.32T + 13T^{2} $
	17	$ 1 - 1.69iT - 17T^{2} $
	19	$ 1 + 7.04iT - 19T^{2} $
	23	$ 1 + 3.12iT - 23T^{2} $
	29	$ 1 - 1.54T + 29T^{2} $
	31	$ 1 + 8.30iT - 31T^{2} $
	37	$ 1 + 4.66iT - 37T^{2} $
	41	$ 1 - 7.73iT - 41T^{2} $
	43	$ 1 - 3.95iT - 43T^{2} $
	47	$ 1 - 6iT - 47T^{2} $
	53	$ 1 - 8.24iT - 53T^{2} $
	59	$ 1 - 11.9T + 59T^{2} $
	61	$ 1 - 8.10T + 61T^{2} $
	67	$ 1 + 6.24T + 67T^{2} $
	71	$ 1 + 12.2iT - 71T^{2} $
	73	$ 1 + 3.08iT - 73T^{2} $
	79	$ 1 - 12.4T + 79T^{2} $
	83	$ 1 + 4.71iT - 83T^{2} $
	89	$ 1 + 16.6T + 89T^{2} $
	97	$ 1 + 13.3T + 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

−8.559265819530087211100029781322, −7.904024132828787331343861794117, −7.14622931142315751394809937059, −6.30974419787907786704196805414, −5.30215273185009316684525105932, −4.51799917464475203238237895549, −4.31756285043683327286969381693, −2.72635489573020620194601770187, −1.77353539771034145792166302182, −0.77131389546558420690160240205, 1.19826718851821747748104164374, 2.18018774917416114806963016768, 3.31464885647746528500727501516, 3.94570305511394097409104244136, 5.11737731406286715163862818677, 5.65507844009595802964616520789, 6.77314763134082511865433207179, 7.13444881332669220639219705353, 8.254192759282531533467104100442, 8.491978372856094870018480473118

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.190455927\)
\(L(\frac12)\)	\(\approx\)	\(2.190455927\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more