LMFDB - L-function 2-6048-12.11-c1-0-33

Dirichlet series

L(s) = 1

− 0.317i·5^-s − i·7^-s + 4.87·11^-s − 2.44·13^-s − 0.317i·17^-s + 0.449i·19^-s − 2.82·23^-s + 4.89·25^-s + 7.70i·29^-s + 4.44i·31^-s − 0.317·35^-s − 7·37^-s + 5.97i·41^-s + 10.3i·43^-s − 3.92·47^-s + ⋯

L(s) = 1

− 0.142i·5^-s − 0.377i·7^-s + 1.47·11^-s − 0.679·13^-s − 0.0770i·17^-s + 0.103i·19^-s − 0.589·23^-s + 0.979·25^-s + 1.43i·29^-s + 0.799i·31^-s − 0.0537·35^-s − 1.15·37^-s + 0.933i·41^-s + 1.57i·43^-s − 0.572·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.857649949$
$L(\frac12)$	$\approx$	$1.857649949$
$L(\frac{3}{2})$		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 $
	7	$ 1 + iT $
good	5	$ 1 + 0.317iT - 5T^{2} $
	11	$ 1 - 4.87T + 11T^{2} $
	13	$ 1 + 2.44T + 13T^{2} $
	17	$ 1 + 0.317iT - 17T^{2} $
	19	$ 1 - 0.449iT - 19T^{2} $
	23	$ 1 + 2.82T + 23T^{2} $
	29	$ 1 - 7.70iT - 29T^{2} $
	31	$ 1 - 4.44iT - 31T^{2} $
	37	$ 1 + 7T + 37T^{2} $
	41	$ 1 - 5.97iT - 41T^{2} $
	43	$ 1 - 10.3iT - 43T^{2} $
	47	$ 1 + 3.92T + 47T^{2} $
	53	$ 1 + 0.635iT - 53T^{2} $
	59	$ 1 - 11.4T + 59T^{2} $
	61	$ 1 - 9.34T + 61T^{2} $
	67	$ 1 + 8iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 3.10T + 73T^{2} $
	79	$ 1 - 8.55iT - 79T^{2} $
	83	$ 1 + 10.2T + 83T^{2} $
	89	$ 1 + 14.1iT - 89T^{2} $
	97	$ 1 - 12.2T + 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

−8.326877613460760039943885327644, −7.23786123676785579543620591051, −6.85108494761317547920514455321, −6.18038141401677628816405673133, −5.13802773802446783326470993454, −4.60634480901555214008836653305, −3.70640182948598609298679444543, −3.04338710699255371533246046680, −1.79573951910634519634507928737, −1.00396151521587073661790763150, 0.54378475954140263886431729626, 1.83312004451942824902122255991, 2.55513351075288154513406793566, 3.69798953415648458570642016016, 4.18222648238256787625575381062, 5.19559152802112171233755828042, 5.84378244811889077046147609868, 6.72071528993584967940918879640, 7.07506178211160351261941287207, 8.059838755896808069250910301273

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.857649949\)
\(L(\frac12)\)	\(\approx\)	\(1.857649949\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more