LMFDB - L-function 2-3168-44.43-c1-0-9

Dirichlet series

L(s) = 1

− 1.86·5^-s − 3.91·7^-s + (−3.30 + 0.269i)11^-s − 3.53i·13^-s − 1.77i·17^-s − 3.02·19^-s − 7.41i·23^-s − 1.50·25^-s − 5.55i·29^-s + 9.85i·31^-s + 7.32·35^-s + 9.85·37^-s + 8.38i·41^-s + 2.26·43^-s − 3.13i·47^-s + ⋯

L(s) = 1

− 0.836·5^-s − 1.48·7^-s + (−0.996 + 0.0813i)11^-s − 0.980i·13^-s − 0.429i·17^-s − 0.694·19^-s − 1.54i·23^-s − 0.300·25^-s − 1.03i·29^-s + 1.76i·31^-s + 1.23·35^-s + 1.62·37^-s + 1.30i·41^-s + 0.345·43^-s − 0.456i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3168$ = $2^{5} \cdot 3^{2} \cdot 11$
Sign:	$0.647 - 0.762i$
Analytic conductor:	$25.2966$
Root analytic conductor:	$5.02957$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3168} (703, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3168,\ (\ :1/2),\ 0.647 - 0.762i)$

Particular Values

$L(1)$	$\approx$	$0.5084041740$
$L(\frac12)$	$\approx$	$0.5084041740$
$L(\frac{3}{2})$		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 + (3.30 - 0.269i)T $
good	5	$ 1 + 1.86T + 5T^{2} $
	7	$ 1 + 3.91T + 7T^{2} $
	13	$ 1 + 3.53iT - 13T^{2} $
	17	$ 1 + 1.77iT - 17T^{2} $
	19	$ 1 + 3.02T + 19T^{2} $
	23	$ 1 + 7.41iT - 23T^{2} $
	29	$ 1 + 5.55iT - 29T^{2} $
	31	$ 1 - 9.85iT - 31T^{2} $
	37	$ 1 - 9.85T + 37T^{2} $
	41	$ 1 - 8.38iT - 41T^{2} $
	43	$ 1 - 2.26T + 43T^{2} $
	47	$ 1 + 3.13iT - 47T^{2} $
	53	$ 1 + 8.13T + 53T^{2} $
	59	$ 1 - 10.5iT - 59T^{2} $
	61	$ 1 + 8.82iT - 61T^{2} $
	67	$ 1 - 4.84iT - 67T^{2} $
	71	$ 1 + 0.607iT - 71T^{2} $
	73	$ 1 - 13.8iT - 73T^{2} $
	79	$ 1 - 3.15T + 79T^{2} $
	83	$ 1 + 6.37T + 83T^{2} $
	89	$ 1 + 15.9T + 89T^{2} $
	97	$ 1 - 9.34T + 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

−8.593072077767754388890867733502, −8.072724827814161311463258721336, −7.33106990401696834516079313055, −6.50453232981345512765071033457, −5.89606218387113773191233054580, −4.84036907081070278212431435227, −4.06327833369204491535835198567, −3.03693368062869330713305184102, −2.60784014300265284691899278510, −0.61295540975632516927818136650, 0.26765941639822051758075183000, 2.01157935010388606808312781227, 3.08775943037852156378659920763, 3.79920862240735355962840118311, 4.50854225508140054793046028915, 5.73591938026435832444469385590, 6.24850668027973228295009528190, 7.22217935327642451948488750124, 7.69425391614095451849955360966, 8.549869773408317447438273726735

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.5084041740\)
\(L(\frac12)\)	\(\approx\)	\(0.5084041740\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more