LMFDB - L-function 2-1152-8.5-c3-0-27

Dirichlet series

L(s) = 1

− 12i·5^-s − 32·7^-s + 8i·11^-s + 20i·13^-s + 98·17^-s + 88i·19^-s − 32·23^-s − 19·25^-s + 172i·29^-s − 256·31^-s + 384i·35^-s + 92i·37^-s + 102·41^-s − 296i·43^-s + 320·47^-s + ⋯

L(s) = 1

− 1.07i·5^-s − 1.72·7^-s + 0.219i·11^-s + 0.426i·13^-s + 1.39·17^-s + 1.06i·19^-s − 0.290·23^-s − 0.151·25^-s + 1.10i·29^-s − 1.48·31^-s + 1.85i·35^-s + 0.408i·37^-s + 0.388·41^-s − 1.04i·43^-s + 0.993·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$67.9702$
Root analytic conductor:	$8.24440$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :3/2),\ 0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$1.352819252$
$L(\frac12)$	$\approx$	$1.352819252$
$L(\frac{5}{2})$		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 $
good	5	$ 1 + 12iT - 125T^{2} $
	7	$ 1 + 32T + 343T^{2} $
	11	$ 1 - 8iT - 1.33e3T^{2} $
	13	$ 1 - 20iT - 2.19e3T^{2} $
	17	$ 1 - 98T + 4.91e3T^{2} $
	19	$ 1 - 88iT - 6.85e3T^{2} $
	23	$ 1 + 32T + 1.21e4T^{2} $
	29	$ 1 - 172iT - 2.43e4T^{2} $
	31	$ 1 + 256T + 2.97e4T^{2} $
	37	$ 1 - 92iT - 5.06e4T^{2} $
	41	$ 1 - 102T + 6.89e4T^{2} $
	43	$ 1 + 296iT - 7.95e4T^{2} $
	47	$ 1 - 320T + 1.03e5T^{2} $
	53	$ 1 + 76iT - 1.48e5T^{2} $
	59	$ 1 + 408iT - 2.05e5T^{2} $
	61	$ 1 + 636iT - 2.26e5T^{2} $
	67	$ 1 + 552iT - 3.00e5T^{2} $
	71	$ 1 - 416T + 3.57e5T^{2} $
	73	$ 1 + 138T + 3.89e5T^{2} $
	79	$ 1 + 64T + 4.93e5T^{2} $
	83	$ 1 - 392iT - 5.71e5T^{2} $
	89	$ 1 + 582T + 7.04e5T^{2} $
	97	$ 1 - 238T + 9.12e5T^{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

−9.392656189466090410047124048107, −8.674862319785661612836621078538, −7.66418723613323904447223764772, −6.79746673901580801844728352654, −5.85967078359538650728688935205, −5.17957456048342191021504473275, −3.88892057072972161905692027248, −3.25915613963670719844321856028, −1.74096922812398386298285286014, −0.52020198516688606631277704698, 0.67341887266093627643859601061, 2.59077756564567735159746202525, 3.15367373787743586084644408835, 4.00312819470748420122643950859, 5.60699025300210885792316331158, 6.15714614460348207807664741314, 7.07104639519245952422796475666, 7.58914452445000718275572054090, 8.890809329735441799240492357002, 9.666664658809018361517647869113

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.352819252\)
\(L(\frac12)\)	\(\approx\)	\(1.352819252\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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