LMFDB - L-function 2-6120-17.16-c1-0-45

Dirichlet series

L(s) = 1

− i·5^-s − 0.489i·7^-s − 0.489i·11^-s + 0.292·13^-s + (−2.19 + 3.48i)17^-s − 7.17·19^-s + 2.29i·23^-s − 25^-s + 1.51i·29^-s − 4.68i·31^-s − 0.489·35^-s + 1.51i·37^-s + 7.86i·41^-s + 9.95·43^-s + 13.5·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 0.184i·7^-s − 0.147i·11^-s + 0.0811·13^-s + (−0.532 + 0.846i)17^-s − 1.64·19^-s + 0.478i·23^-s − 0.200·25^-s + 0.280i·29^-s − 0.841i·31^-s − 0.0827·35^-s + 0.248i·37^-s + 1.22i·41^-s + 1.51·43^-s + 1.97·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6120$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 17$
Sign:	$0.846 + 0.532i$
Analytic conductor:	$48.8684$
Root analytic conductor:	$6.99059$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6120} (1801, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6120,\ (\ :1/2),\ 0.846 + 0.532i)$

Particular Values

$L(1)$	$\approx$	$1.658300399$
$L(\frac12)$	$\approx$	$1.658300399$
$L(\frac{3}{2})$		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 + iT $
	17	$ 1 + (2.19 - 3.48i)T $
good	7	$ 1 + 0.489iT - 7T^{2} $
	11	$ 1 + 0.489iT - 11T^{2} $
	13	$ 1 - 0.292T + 13T^{2} $
	19	$ 1 + 7.17T + 19T^{2} $
	23	$ 1 - 2.29iT - 23T^{2} $
	29	$ 1 - 1.51iT - 29T^{2} $
	31	$ 1 + 4.68iT - 31T^{2} $
	37	$ 1 - 1.51iT - 37T^{2} $
	41	$ 1 - 7.86iT - 41T^{2} $
	43	$ 1 - 9.95T + 43T^{2} $
	47	$ 1 - 13.5T + 47T^{2} $
	53	$ 1 + 1.80T + 53T^{2} $
	59	$ 1 - 7.07T + 59T^{2} $
	61	$ 1 + 5.70iT - 61T^{2} $
	67	$ 1 - 7.27T + 67T^{2} $
	71	$ 1 - 0.585iT - 71T^{2} $
	73	$ 1 + 16.8iT - 73T^{2} $
	79	$ 1 + 15.9iT - 79T^{2} $
	83	$ 1 - 10.3T + 83T^{2} $
	89	$ 1 - 4.68T + 89T^{2} $
	97	$ 1 - 3.95iT - 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

−8.007064335075448980749951120372, −7.41501938213716902610375169277, −6.38398946142241896259946840457, −6.04632002546085230151611608308, −5.08445865238494763986500161557, −4.23835399643771627919248962002, −3.81367570666025512299658884843, −2.56705030079380637250679071922, −1.78630415111860286542859030890, −0.59778565897287705433973027982, 0.73542659849945289134755050936, 2.26750431824503419211150423726, 2.56169827577738180335151126682, 3.88174848661140195149499417311, 4.32980926577032009770173838860, 5.37142278222587794695119173880, 5.98511578217747247752329019223, 6.92977140801263009584416800744, 7.14523516236785133112943586421, 8.236615322691790416040992617494

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.658300399\)
\(L(\frac12)\)	\(\approx\)	\(1.658300399\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more