LMFDB - L-function 2-6048-24.11-c1-0-65

Dirichlet series

L(s) = 1

− 1.27·5^-s − i·7^-s + 2.84i·11^-s + 0.223i·13^-s − 0.397i·17^-s + 4.64·19^-s − 7.36·23^-s − 3.36·25^-s + 10.6·29^-s − 7.67i·31^-s + 1.27i·35^-s − 4.74i·37^-s + 1.97i·41^-s − 7.62·43^-s − 11.0·47^-s + ⋯

L(s) = 1

− 0.571·5^-s − 0.377i·7^-s + 0.858i·11^-s + 0.0620i·13^-s − 0.0964i·17^-s + 1.06·19^-s − 1.53·23^-s − 0.673·25^-s + 1.97·29^-s − 1.37i·31^-s + 0.216i·35^-s − 0.780i·37^-s + 0.308i·41^-s − 1.16·43^-s − 1.61·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0574 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0574 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.0574 + 0.998i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (5615, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ -0.0574 + 0.998i)$

Particular Values

$L(1)$	$\approx$	$1.040279889$
$L(\frac12)$	$\approx$	$1.040279889$
$L(\frac{3}{2})$		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 + 1.27T + 5T^{2} $
	11	$ 1 - 2.84iT - 11T^{2} $
	13	$ 1 - 0.223iT - 13T^{2} $
	17	$ 1 + 0.397iT - 17T^{2} $
	19	$ 1 - 4.64T + 19T^{2} $
	23	$ 1 + 7.36T + 23T^{2} $
	29	$ 1 - 10.6T + 29T^{2} $
	31	$ 1 + 7.67iT - 31T^{2} $
	37	$ 1 + 4.74iT - 37T^{2} $
	41	$ 1 - 1.97iT - 41T^{2} $
	43	$ 1 + 7.62T + 43T^{2} $
	47	$ 1 + 11.0T + 47T^{2} $
	53	$ 1 + 0.295T + 53T^{2} $
	59	$ 1 - 7.25iT - 59T^{2} $
	61	$ 1 - 9.45iT - 61T^{2} $
	67	$ 1 - 3.52T + 67T^{2} $
	71	$ 1 - 8.37T + 71T^{2} $
	73	$ 1 - 10.6T + 73T^{2} $
	79	$ 1 + 5.22iT - 79T^{2} $
	83	$ 1 + 9.11iT - 83T^{2} $
	89	$ 1 - 8.94iT - 89T^{2} $
	97	$ 1 - 0.228T + 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

−7.926791655194202771884609152693, −7.26955496860845326176030030681, −6.55978099026365103984114232700, −5.79093886841293946581859405245, −4.85308698504200538002327762157, −4.23599791452400809828081531045, −3.55416018536895139610445227224, −2.54965714580771312410270745317, −1.57262730493874515672700437976, −0.31058429216505127774617688533, 0.966833375167859802316198198063, 2.11437499139526113947796320718, 3.25460582895426218266128027842, 3.59922769702337509688170733802, 4.78411436434469693522550744598, 5.27295757628931176944681574818, 6.31016968824564963267404319971, 6.65003585125937161507119861622, 7.82065843262633332871750848497, 8.192180093866048794693635023591

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.040279889\)
\(L(\frac12)\)	\(\approx\)	\(1.040279889\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more