LMFDB - L-function 2-6048-24.11-c1-0-28

Dirichlet series

L(s) = 1

− 2.21·5^-s − i·7^-s + 2.42i·11^-s + 1.11i·13^-s − 0.701i·17^-s − 0.938·19^-s + 4.30·23^-s − 0.102·25^-s + 4.26·29^-s − 7.02i·31^-s + 2.21i·35^-s − 3.12i·37^-s − 0.157i·41^-s − 7.08·43^-s − 0.867·47^-s + ⋯

L(s) = 1

− 0.989·5^-s − 0.377i·7^-s + 0.730i·11^-s + 0.309i·13^-s − 0.170i·17^-s − 0.215·19^-s + 0.897·23^-s − 0.0204·25^-s + 0.791·29^-s − 1.26i·31^-s + 0.374i·35^-s − 0.513i·37^-s − 0.0246i·41^-s − 1.08·43^-s − 0.126·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.727 - 0.686i$
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:	$(2,\ 6048,\ (\ :1/2),\ 0.727 - 0.686i)$

Particular Values

$L(1)$	$\approx$	$1.220903645$
$L(\frac12)$	$\approx$	$1.220903645$
$L(\frac{3}{2})$		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 + 2.21T + 5T^{2} $
	11	$ 1 - 2.42iT - 11T^{2} $
	13	$ 1 - 1.11iT - 13T^{2} $
	17	$ 1 + 0.701iT - 17T^{2} $
	19	$ 1 + 0.938T + 19T^{2} $
	23	$ 1 - 4.30T + 23T^{2} $
	29	$ 1 - 4.26T + 29T^{2} $
	31	$ 1 + 7.02iT - 31T^{2} $
	37	$ 1 + 3.12iT - 37T^{2} $
	41	$ 1 + 0.157iT - 41T^{2} $
	43	$ 1 + 7.08T + 43T^{2} $
	47	$ 1 + 0.867T + 47T^{2} $
	53	$ 1 - 3.91T + 53T^{2} $
	59	$ 1 - 7.28iT - 59T^{2} $
	61	$ 1 + 4.57iT - 61T^{2} $
	67	$ 1 + 15.1T + 67T^{2} $
	71	$ 1 + 6.93T + 71T^{2} $
	73	$ 1 - 7.02T + 73T^{2} $
	79	$ 1 - 6.65iT - 79T^{2} $
	83	$ 1 - 8.41iT - 83T^{2} $
	89	$ 1 - 7.34iT - 89T^{2} $
	97	$ 1 - 7.99T + 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

−8.023587435130852096366905495140, −7.45733318370124347887387103404, −6.91973933565898856131698288819, −6.14641540491835142708237354714, −5.11489210390606931807251937937, −4.42295200465695761333512151915, −3.88158006003341174392272502017, −2.97524822454117913474147768559, −1.96913182368824417980503126783, −0.74567762431460259296341890467, 0.45502536183360928145579912174, 1.64102522064385391604740896746, 3.00074963283883144076783486291, 3.34271118428049085010247869062, 4.39419436967559148831533852543, 5.02452416753042013859811436066, 5.88277658121896838080356983303, 6.61461207949011650005726323434, 7.33834794214976718568485363278, 8.075223505880459152386696259892

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.220903645\)
\(L(\frac12)\)	\(\approx\)	\(1.220903645\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more