LMFDB - L-function 2-864-24.11-c3-0-0

Dirichlet series

L(s) = 1

− 12.8·5^-s − 12.3i·7^-s + 24.6i·11^-s − 2.63i·13^-s − 93.1i·17^-s + 133.·19^-s + 17.4·23^-s + 39.2·25^-s − 181.·29^-s − 304. i·31^-s + 157. i·35^-s + 81.2i·37^-s + 314. i·41^-s − 333.·43^-s + 52.6·47^-s + ⋯

L(s) = 1

− 1.14·5^-s − 0.664i·7^-s + 0.676i·11^-s − 0.0563i·13^-s − 1.32i·17^-s + 1.61·19^-s + 0.158·23^-s + 0.314·25^-s − 1.15·29^-s − 1.76i·31^-s + 0.762i·35^-s + 0.360i·37^-s + 1.19i·41^-s − 1.18·43^-s + 0.163·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$-0.699 - 0.714i$
Analytic conductor:	$50.9776$
Root analytic conductor:	$7.13986$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :3/2),\ -0.699 - 0.714i)$

Particular Values

$L(2)$	$\approx$	$0.2802015424$
$L(\frac12)$	$\approx$	$0.2802015424$
$L(\frac{5}{2})$		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 + 12.8T + 125T^{2} $
	7	$ 1 + 12.3iT - 343T^{2} $
	11	$ 1 - 24.6iT - 1.33e3T^{2} $
	13	$ 1 + 2.63iT - 2.19e3T^{2} $
	17	$ 1 + 93.1iT - 4.91e3T^{2} $
	19	$ 1 - 133.T + 6.85e3T^{2} $
	23	$ 1 - 17.4T + 1.21e4T^{2} $
	29	$ 1 + 181.T + 2.43e4T^{2} $
	31	$ 1 + 304. iT - 2.97e4T^{2} $
	37	$ 1 - 81.2iT - 5.06e4T^{2} $
	41	$ 1 - 314. iT - 6.89e4T^{2} $
	43	$ 1 + 333.T + 7.95e4T^{2} $
	47	$ 1 - 52.6T + 1.03e5T^{2} $
	53	$ 1 + 227.T + 1.48e5T^{2} $
	59	$ 1 - 256. iT - 2.05e5T^{2} $
	61	$ 1 - 808. iT - 2.26e5T^{2} $
	67	$ 1 - 110.T + 3.00e5T^{2} $
	71	$ 1 + 582.T + 3.57e5T^{2} $
	73	$ 1 + 290.T + 3.89e5T^{2} $
	79	$ 1 - 485. iT - 4.93e5T^{2} $
	83	$ 1 - 1.13e3iT - 5.71e5T^{2} $
	89	$ 1 - 1.01e3iT - 7.04e5T^{2} $
	97	$ 1 + 1.26e3T + 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

−9.890470442391402714976176269620, −9.465335827638328044734398654965, −8.175710784441697271762133145454, −7.40998427295546841322138041707, −7.09311935223885792221515779855, −5.60820485721007713575024386286, −4.60286269545455265982109175733, −3.81960242895171751128944125654, −2.78052896431190826877039023060, −1.09755345585016767560927004724, 0.084644274034507433093661210581, 1.59117507208680745508923023720, 3.18531703347274825265490869288, 3.78023934176963471088874246425, 5.07741656543527241811210899348, 5.87458520285266986357071167949, 7.01300337180359119464855541151, 7.84870103358659834716461509662, 8.557059242588059011871664968215, 9.285484840387034911296597229697

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.2802015424\)
\(L(\frac12)\)	\(\approx\)	\(0.2802015424\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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