LMFDB - L-function 2-2520-5.4-c1-0-15

Dirichlet series

L(s) = 1

+ (0.311 + 2.21i)5^-s + i·7^-s + 3.80·11^-s − 0.622i·13^-s + 4.42i·17^-s − 0.622·19^-s + 2.62i·23^-s + (−4.80 + 1.37i)25^-s + 9.61·29^-s − 0.622·31^-s + (−2.21 + 0.311i)35^-s − 1.24i·37^-s − 4.62·41^-s + 4.85i·43^-s − 11.6i·47^-s + ⋯

L(s) = 1

+ (0.139 + 0.990i)5^-s + 0.377i·7^-s + 1.14·11^-s − 0.172i·13^-s + 1.07i·17^-s − 0.142·19^-s + 0.546i·23^-s + (−0.961 + 0.275i)25^-s + 1.78·29^-s − 0.111·31^-s + (−0.374 + 0.0525i)35^-s − 0.204i·37^-s − 0.721·41^-s + 0.740i·43^-s − 1.69i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2520$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
Sign:	$-0.139 - 0.990i$
Analytic conductor:	$20.1223$
Root analytic conductor:	$4.48578$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2520} (1009, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2520,\ (\ :1/2),\ -0.139 - 0.990i)$

Particular Values

$L(1)$	$\approx$	$1.803526155$
$L(\frac12)$	$\approx$	$1.803526155$
$L(\frac{3}{2})$		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 $
	5	$ 1 + (-0.311 - 2.21i)T $
	7	$ 1 - iT $
good	11	$ 1 - 3.80T + 11T^{2} $
	13	$ 1 + 0.622iT - 13T^{2} $
	17	$ 1 - 4.42iT - 17T^{2} $
	19	$ 1 + 0.622T + 19T^{2} $
	23	$ 1 - 2.62iT - 23T^{2} $
	29	$ 1 - 9.61T + 29T^{2} $
	31	$ 1 + 0.622T + 31T^{2} $
	37	$ 1 + 1.24iT - 37T^{2} $
	41	$ 1 + 4.62T + 41T^{2} $
	43	$ 1 - 4.85iT - 43T^{2} $
	47	$ 1 + 11.6iT - 47T^{2} $
	53	$ 1 - 13.4iT - 53T^{2} $
	59	$ 1 - 11.6T + 59T^{2} $
	61	$ 1 + 8.10T + 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 2.56T + 71T^{2} $
	73	$ 1 + 10.9iT - 73T^{2} $
	79	$ 1 - 6.75T + 79T^{2} $
	83	$ 1 - 11.6iT - 83T^{2} $
	89	$ 1 + 8.23T + 89T^{2} $
	97	$ 1 - 4.23iT - 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

−9.085655711315005492530758519934, −8.399191590576381358965381121435, −7.55565236226634315958164888889, −6.59932029969779558703552547355, −6.28224902130954703856177881159, −5.32790725550374475377381207681, −4.16343818137939350829768178121, −3.43613842668041756987824931893, −2.46979072961273578477879157300, −1.39044425112086682275302086815, 0.64126499306672039365435471385, 1.62319907664846967585794287814, 2.90864718630323610424312967692, 4.08723990834652325301504129882, 4.64396588908693837579906487429, 5.48561023140235625849655216476, 6.52192095627603730116034493254, 7.02317247799003972118846883533, 8.169726443038387390245182239683, 8.656764878190742017468808883079

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.803526155\)
\(L(\frac12)\)	\(\approx\)	\(1.803526155\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more