LMFDB - L-function 2-864-24.11-c3-0-6

Dirichlet series

L(s) = 1

− 13.3·5^-s + 1.40i·7^-s − 17.6i·11^-s − 70.2i·13^-s + 79.0i·17^-s − 107.·19^-s − 41.7·23^-s + 52.4·25^-s + 282.·29^-s − 84.6i·31^-s − 18.6i·35^-s − 292. i·37^-s + 108. i·41^-s − 29.7·43^-s − 206.·47^-s + ⋯

L(s) = 1

− 1.19·5^-s + 0.0757i·7^-s − 0.484i·11^-s − 1.49i·13^-s + 1.12i·17^-s − 1.29·19^-s − 0.378·23^-s + 0.419·25^-s + 1.81·29^-s − 0.490i·31^-s − 0.0902i·35^-s − 1.29i·37^-s + 0.415i·41^-s − 0.105·43^-s − 0.641·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.273 - 0.961i$
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:	$(2,\ 864,\ (\ :3/2),\ 0.273 - 0.961i)$

Particular Values

$L(2)$	$\approx$	$0.8090717236$
$L(\frac12)$	$\approx$	$0.8090717236$
$L(\frac{5}{2})$		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 + 13.3T + 125T^{2} $
	7	$ 1 - 1.40iT - 343T^{2} $
	11	$ 1 + 17.6iT - 1.33e3T^{2} $
	13	$ 1 + 70.2iT - 2.19e3T^{2} $
	17	$ 1 - 79.0iT - 4.91e3T^{2} $
	19	$ 1 + 107.T + 6.85e3T^{2} $
	23	$ 1 + 41.7T + 1.21e4T^{2} $
	29	$ 1 - 282.T + 2.43e4T^{2} $
	31	$ 1 + 84.6iT - 2.97e4T^{2} $
	37	$ 1 + 292. iT - 5.06e4T^{2} $
	41	$ 1 - 108. iT - 6.89e4T^{2} $
	43	$ 1 + 29.7T + 7.95e4T^{2} $
	47	$ 1 + 206.T + 1.03e5T^{2} $
	53	$ 1 + 207.T + 1.48e5T^{2} $
	59	$ 1 - 618. iT - 2.05e5T^{2} $
	61	$ 1 - 667. iT - 2.26e5T^{2} $
	67	$ 1 + 848.T + 3.00e5T^{2} $
	71	$ 1 - 535.T + 3.57e5T^{2} $
	73	$ 1 - 397.T + 3.89e5T^{2} $
	79	$ 1 - 1.19e3iT - 4.93e5T^{2} $
	83	$ 1 - 636. iT - 5.71e5T^{2} $
	89	$ 1 + 179. iT - 7.04e5T^{2} $
	97	$ 1 - 550.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

−10.26653894256591669688833244720, −8.806565202971469144923934168134, −8.201603239350749929426454627582, −7.66616158577319218144933420760, −6.45196754078259034616355999952, −5.64596666029284005301027585022, −4.38625015703669265854652550540, −3.66092998779805106986555794153, −2.56097474249551958393144271455, −0.835943702123356769108405022820, 0.28635305745905539563356903102, 1.88646133539899118662444546125, 3.19735715069696695357230018926, 4.36995528935512542940236543840, 4.75037263472193630977281116955, 6.44369401503080003225534311311, 6.96725147434814873918195680785, 7.951035460525964820203846290311, 8.665659096292574916487264476021, 9.559670591605492404679038125521

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8090717236\)
\(L(\frac12)\)	\(\approx\)	\(0.8090717236\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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