LMFDB - L-function 2-24e2-8.5-c3-0-5

Dirichlet series

L(s) = 1

− 13.8i·5^-s + 3.46·7^-s + 48i·11^-s + 20.7i·13^-s − 96·17^-s + 40i·19^-s − 110.·23^-s − 66.9·25^-s − 13.8i·29^-s + 204.·31^-s − 47.9i·35^-s + 297. i·37^-s + 288·41^-s + 152i·43^-s + 554.·47^-s + ⋯

L(s) = 1

− 1.23i·5^-s + 0.187·7^-s + 1.31i·11^-s + 0.443i·13^-s − 1.36·17^-s + 0.482i·19^-s − 1.00·23^-s − 0.535·25^-s − 0.0887i·29^-s + 1.18·31^-s − 0.231i·35^-s + 1.32i·37^-s + 1.09·41^-s + 0.539i·43^-s + 1.72·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$576$ = $2^{6} \cdot 3^{2}$
Sign:	$0.258 - 0.965i$
Analytic conductor:	$33.9851$
Root analytic conductor:	$5.82967$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{576} (289, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 576,\ (\ :3/2),\ 0.258 - 0.965i)$

Particular Values

$L(2)$	$\approx$	$1.263682639$
$L(\frac12)$	$\approx$	$1.263682639$
$L(\frac{5}{2})$		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 + 13.8iT - 125T^{2} $
	7	$ 1 - 3.46T + 343T^{2} $
	11	$ 1 - 48iT - 1.33e3T^{2} $
	13	$ 1 - 20.7iT - 2.19e3T^{2} $
	17	$ 1 + 96T + 4.91e3T^{2} $
	19	$ 1 - 40iT - 6.85e3T^{2} $
	23	$ 1 + 110.T + 1.21e4T^{2} $
	29	$ 1 + 13.8iT - 2.43e4T^{2} $
	31	$ 1 - 204.T + 2.97e4T^{2} $
	37	$ 1 - 297. iT - 5.06e4T^{2} $
	41	$ 1 - 288T + 6.89e4T^{2} $
	43	$ 1 - 152iT - 7.95e4T^{2} $
	47	$ 1 - 554.T + 1.03e5T^{2} $
	53	$ 1 - 180. iT - 1.48e5T^{2} $
	59	$ 1 + 480iT - 2.05e5T^{2} $
	61	$ 1 - 755. iT - 2.26e5T^{2} $
	67	$ 1 - 848iT - 3.00e5T^{2} $
	71	$ 1 + 886.T + 3.57e5T^{2} $
	73	$ 1 + 538T + 3.89e5T^{2} $
	79	$ 1 - 1.00e3T + 4.93e5T^{2} $
	83	$ 1 - 432iT - 5.71e5T^{2} $
	89	$ 1 + 1.34e3T + 7.04e5T^{2} $
	97	$ 1 + 590T + 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

−10.33975862892823934405088782214, −9.554826105863832336754782544830, −8.765054864115026961828501000179, −7.983363675675960617107866944439, −6.93297668198878902858378799583, −5.87630813846473721930957690686, −4.57288351966548682677435643443, −4.33797200267635410905716941495, −2.35807579429580258950831657477, −1.26034651133895404860356668779, 0.38825598537951438965242996898, 2.29572692098964820748320495240, 3.20450728824967372718284750350, 4.33429475238747031501053304008, 5.76969852119645143867107601574, 6.45688701946566511894266438976, 7.40107591493420919535028986484, 8.351160493966307508768928416231, 9.221655842316071477545785898266, 10.43285320840816177443204840015

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.263682639\)
\(L(\frac12)\)	\(\approx\)	\(1.263682639\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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