LMFDB - L-function 2-6048-12.11-c1-0-68

Dirichlet series

L(s) = 1

− 0.517i·5^-s + i·7^-s − 5.63·11^-s + 1.91·13^-s + 1.29i·17^-s + 4.23i·19^-s − 4.38·23^-s + 4.73·25^-s − 3.70i·29^-s + 5.11i·31^-s + 0.517·35^-s + 2.18·37^-s + 1.88i·41^-s − 5.37i·43^-s + 8.32·47^-s + ⋯

L(s) = 1

− 0.231i·5^-s + 0.377i·7^-s − 1.69·11^-s + 0.531·13^-s + 0.314i·17^-s + 0.971i·19^-s − 0.913·23^-s + 0.946·25^-s − 0.687i·29^-s + 0.918i·31^-s + 0.0874·35^-s + 0.358·37^-s + 0.294i·41^-s − 0.820i·43^-s + 1.21·47^-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

$L(1)$	$\approx$	$0.4445294439$
$L(\frac12)$	$\approx$	$0.4445294439$
$L(\frac{3}{2})$		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 + 0.517iT - 5T^{2} $
	11	$ 1 + 5.63T + 11T^{2} $
	13	$ 1 - 1.91T + 13T^{2} $
	17	$ 1 - 1.29iT - 17T^{2} $
	19	$ 1 - 4.23iT - 19T^{2} $
	23	$ 1 + 4.38T + 23T^{2} $
	29	$ 1 + 3.70iT - 29T^{2} $
	31	$ 1 - 5.11iT - 31T^{2} $
	37	$ 1 - 2.18T + 37T^{2} $
	41	$ 1 - 1.88iT - 41T^{2} $
	43	$ 1 + 5.37iT - 43T^{2} $
	47	$ 1 - 8.32T + 47T^{2} $
	53	$ 1 + 5.53iT - 53T^{2} $
	59	$ 1 - 2.54T + 59T^{2} $
	61	$ 1 + 0.0617T + 61T^{2} $
	67	$ 1 + 7.31iT - 67T^{2} $
	71	$ 1 + 13.4T + 71T^{2} $
	73	$ 1 + 14.6T + 73T^{2} $
	79	$ 1 + 16.9iT - 79T^{2} $
	83	$ 1 + 8.59T + 83T^{2} $
	89	$ 1 + 1.33iT - 89T^{2} $
	97	$ 1 - 5.23T + 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−7.87499858599900015530119160029, −7.26140628722161556020186119547, −6.18026472438797173612321594602, −5.71356295598202818101767776170, −5.00127392019765020110644337439, −4.19051523673267322356799381432, −3.24880129878797165156143738694, −2.46329843443583406376166281684, −1.52626658171478029599031979934, −0.11950987000577543486535294567, 1.07775470928758121181908565081, 2.47005909837063214453454473324, 2.89852429814616581135566335266, 4.00905497570637354443008705255, 4.73157282929139268921383665794, 5.50626384814095076794373743155, 6.14804417737878274444873919627, 7.14204368453501111265372994403, 7.49969017328417078965036981837, 8.317553605599352512424496734467

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4445294439\)
\(L(\frac12)\)	\(\approx\)	\(0.4445294439\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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