LMFDB - L-function 2-864-24.11-c3-0-43

Dirichlet series

L(s) = 1

− 21.0·5^-s − 25.8i·7^-s − 46.9i·11^-s − 55.4i·13^-s − 64.8i·17^-s + 51.0·19^-s − 60.4·23^-s + 316.·25^-s − 90.5·29^-s + 8.35i·31^-s + 542. i·35^-s − 228. i·37^-s − 239. i·41^-s + 192.·43^-s − 38.1·47^-s + ⋯

L(s) = 1

− 1.88·5^-s − 1.39i·7^-s − 1.28i·11^-s − 1.18i·13^-s − 0.924i·17^-s + 0.616·19^-s − 0.548·23^-s + 2.53·25^-s − 0.579·29^-s + 0.0484i·31^-s + 2.62i·35^-s − 1.01i·37^-s − 0.913i·41^-s + 0.684·43^-s − 0.118·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$-0.829 - 0.558i$
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:	$(2,\ 864,\ (\ :3/2),\ -0.829 - 0.558i)$

Particular Values

$L(2)$	$\approx$	$0.7612643001$
$L(\frac12)$	$\approx$	$0.7612643001$
$L(\frac{5}{2})$		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 + 21.0T + 125T^{2} $
	7	$ 1 + 25.8iT - 343T^{2} $
	11	$ 1 + 46.9iT - 1.33e3T^{2} $
	13	$ 1 + 55.4iT - 2.19e3T^{2} $
	17	$ 1 + 64.8iT - 4.91e3T^{2} $
	19	$ 1 - 51.0T + 6.85e3T^{2} $
	23	$ 1 + 60.4T + 1.21e4T^{2} $
	29	$ 1 + 90.5T + 2.43e4T^{2} $
	31	$ 1 - 8.35iT - 2.97e4T^{2} $
	37	$ 1 + 228. iT - 5.06e4T^{2} $
	41	$ 1 + 239. iT - 6.89e4T^{2} $
	43	$ 1 - 192.T + 7.95e4T^{2} $
	47	$ 1 + 38.1T + 1.03e5T^{2} $
	53	$ 1 - 6.69T + 1.48e5T^{2} $
	59	$ 1 + 5.40iT - 2.05e5T^{2} $
	61	$ 1 + 847. iT - 2.26e5T^{2} $
	67	$ 1 + 992.T + 3.00e5T^{2} $
	71	$ 1 + 494.T + 3.57e5T^{2} $
	73	$ 1 + 395.T + 3.89e5T^{2} $
	79	$ 1 + 42.1iT - 4.93e5T^{2} $
	83	$ 1 + 328. iT - 5.71e5T^{2} $
	89	$ 1 - 1.13e3iT - 7.04e5T^{2} $
	97	$ 1 - 991.T + 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

−9.143950525649357253798544790682, −8.139824363925271207069153227955, −7.61422956264418885565150353469, −7.07634197539708309943091584245, −5.69369409190069118209442628701, −4.55738476910285831395709856306, −3.65811981647020855935418947199, −3.15023171176645699565524372311, −0.75407127658672776293419635582, −0.30739054170250700587164994447, 1.67212888633887536069600831579, 2.94339968577417363972331300211, 4.12014062310124817517961086413, 4.69050656649796753794640472016, 5.97491626605029585245017621309, 7.05481368561931252473531659636, 7.73124738796394369576486645969, 8.584543055471530604335238061480, 9.225115278610879241944291306447, 10.28183946665015918359576192020

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.7612643001\)
\(L(\frac12)\)	\(\approx\)	\(0.7612643001\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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