LMFDB - L-function 2-24e2-3.2-c2-0-3

Dirichlet series

L(s) = 1

− 4.24i·5^-s − 4·7^-s + 16.9i·11^-s − 8·13^-s + 12.7i·17^-s + 16·19^-s + 16.9i·23^-s + 7.00·25^-s + 4.24i·29^-s + 44·31^-s + 16.9i·35^-s + 34·37^-s − 46.6i·41^-s + 40·43^-s + 84.8i·47^-s + ⋯

L(s) = 1

− 0.848i·5^-s − 0.571·7^-s + 1.54i·11^-s − 0.615·13^-s + 0.748i·17^-s + 0.842·19^-s + 0.737i·23^-s + 0.280·25^-s + 0.146i·29^-s + 1.41·31^-s + 0.484i·35^-s + 0.918·37^-s − 1.13i·41^-s + 0.930·43^-s + 1.80i·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

$L(\frac{3}{2})$	$\approx$	$1.392191260$
$L(\frac12)$	$\approx$	$1.392191260$
$L(2)$		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 + 4.24iT - 25T^{2} $
	7	$ 1 + 4T + 49T^{2} $
	11	$ 1 - 16.9iT - 121T^{2} $
	13	$ 1 + 8T + 169T^{2} $
	17	$ 1 - 12.7iT - 289T^{2} $
	19	$ 1 - 16T + 361T^{2} $
	23	$ 1 - 16.9iT - 529T^{2} $
	29	$ 1 - 4.24iT - 841T^{2} $
	31	$ 1 - 44T + 961T^{2} $
	37	$ 1 - 34T + 1.36e3T^{2} $
	41	$ 1 + 46.6iT - 1.68e3T^{2} $
	43	$ 1 - 40T + 1.84e3T^{2} $
	47	$ 1 - 84.8iT - 2.20e3T^{2} $
	53	$ 1 - 38.1iT - 2.80e3T^{2} $
	59	$ 1 - 33.9iT - 3.48e3T^{2} $
	61	$ 1 + 50T + 3.72e3T^{2} $
	67	$ 1 + 8T + 4.48e3T^{2} $
	71	$ 1 - 50.9iT - 5.04e3T^{2} $
	73	$ 1 + 16T + 5.32e3T^{2} $
	79	$ 1 + 76T + 6.24e3T^{2} $
	83	$ 1 - 118. iT - 6.88e3T^{2} $
	89	$ 1 + 12.7iT - 7.92e3T^{2} $
	97	$ 1 - 176T + 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

−10.44882329623251988886633529092, −9.654189517006674460684717817632, −9.129393105507582494888270405018, −7.901414971593580332363917520714, −7.18317636281495382331816379031, −6.05134961276261328388706934089, −4.95170075849796652750007623942, −4.18765788399385224104826914129, −2.70000423826547743465227700644, −1.26417550957377167244564087988, 0.58848311151016488524009744380, 2.71658937507894777929043254397, 3.30162842428833956952982956579, 4.78134354405892140433843986155, 5.99403464681577799305453674611, 6.68016659667006142708363365500, 7.65565122356251904222467093466, 8.628668861269995369771418312991, 9.641766286495501295678462943012, 10.35234643651384293253900603358

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.392191260\)
\(L(\frac12)\)	\(\approx\)	\(1.392191260\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more