LMFDB - L-function 2-24e2-3.2-c2-0-8

Dirichlet series

L(s) = 1

− 1.41i·5^-s − 8·7^-s + 11.3i·11^-s + 8·13^-s − 12.7i·17^-s + 32·19^-s − 33.9i·23^-s + 23·25^-s − 43.8i·29^-s − 40·31^-s + 11.3i·35^-s + 26·37^-s − 66.4i·41^-s + 16·43^-s + 11.3i·47^-s + ⋯

L(s) = 1

− 0.282i·5^-s − 1.14·7^-s + 1.02i·11^-s + 0.615·13^-s − 0.748i·17^-s + 1.68·19^-s − 1.47i·23^-s + 0.920·25^-s − 1.51i·29^-s − 1.29·31^-s + 0.323i·35^-s + 0.702·37^-s − 1.62i·41^-s + 0.372·43^-s + 0.240i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$576$ = $2^{6} \cdot 3^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$15.6948$
Root analytic conductor:	$3.96167$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{576} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 576,\ (\ :1),\ 0.577 + 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.455666051$
$L(\frac12)$	$\approx$	$1.455666051$
$L(2)$		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 $
good	5	$ 1 + 1.41iT - 25T^{2} $
	7	$ 1 + 8T + 49T^{2} $
	11	$ 1 - 11.3iT - 121T^{2} $
	13	$ 1 - 8T + 169T^{2} $
	17	$ 1 + 12.7iT - 289T^{2} $
	19	$ 1 - 32T + 361T^{2} $
	23	$ 1 + 33.9iT - 529T^{2} $
	29	$ 1 + 43.8iT - 841T^{2} $
	31	$ 1 + 40T + 961T^{2} $
	37	$ 1 - 26T + 1.36e3T^{2} $
	41	$ 1 + 66.4iT - 1.68e3T^{2} $
	43	$ 1 - 16T + 1.84e3T^{2} $
	47	$ 1 - 11.3iT - 2.20e3T^{2} $
	53	$ 1 + 32.5iT - 2.80e3T^{2} $
	59	$ 1 - 22.6iT - 3.48e3T^{2} $
	61	$ 1 - 54T + 3.72e3T^{2} $
	67	$ 1 + 80T + 4.48e3T^{2} $
	71	$ 1 - 79.1iT - 5.04e3T^{2} $
	73	$ 1 - 96T + 5.32e3T^{2} $
	79	$ 1 - 104T + 6.24e3T^{2} $
	83	$ 1 + 101. iT - 6.88e3T^{2} $
	89	$ 1 + 77.7iT - 7.92e3T^{2} $
	97	$ 1 + 80T + 9.40e3T^{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

−10.18740867922241108472127756552, −9.540442134613457154532251798078, −8.829658673050180257031899691127, −7.55795371199955245316332923015, −6.83954241802546501518369605901, −5.82273014353618091713079678807, −4.74765162343635167813083583267, −3.60420303233784191357032967189, −2.43863138338158499598562987458, −0.65309059300909970336627435040, 1.18023741532820310309672043739, 3.15887468577234989542557758485, 3.54396501780259619779561294411, 5.30119149549375177619160604510, 6.09761832254823018493161628323, 6.98568704424141534151736360160, 7.966221066749094383374244199874, 9.084701781411869464813862659221, 9.643956578120976376456373787195, 10.77606732635183931991648051948

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.455666051\)
\(L(\frac12)\)	\(\approx\)	\(1.455666051\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more