LMFDB - L-function 2-24e2-24.5-c6-0-12

Dirichlet series

L(s) = 1

− 14.3·5^-s − 532.·7^-s − 1.84e3·11^-s − 1.00e3i·13^-s − 3.71e3i·17^-s + 7.86e3i·19^-s + 1.95e4i·23^-s − 1.54e4·25^-s − 4.25e4·29^-s − 3.91e4·31^-s + 7.65e3·35^-s − 6.17e4i·37^-s + 1.33e4i·41^-s − 5.79e4i·43^-s + 5.19e4i·47^-s + ⋯

L(s) = 1

− 0.115·5^-s − 1.55·7^-s − 1.38·11^-s − 0.457i·13^-s − 0.756i·17^-s + 1.14i·19^-s + 1.60i·23^-s − 0.986·25^-s − 1.74·29^-s − 1.31·31^-s + 0.178·35^-s − 1.22i·37^-s + 0.193i·41^-s − 0.728i·43^-s + 0.500i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(7-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$576$ = $2^{6} \cdot 3^{2}$
Sign:	$0.985 + 0.169i$
Analytic conductor:	$132.511$
Root analytic conductor:	$11.5113$
Motivic weight:	$6$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{576} (161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 576,\ (\ :3),\ 0.985 + 0.169i)$

Particular Values

$L(\frac{7}{2})$	$\approx$	$0.4139697591$
$L(\frac12)$	$\approx$	$0.4139697591$
$L(4)$		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 + 14.3T + 1.56e4T^{2} $
	7	$ 1 + 532.T + 1.17e5T^{2} $
	11	$ 1 + 1.84e3T + 1.77e6T^{2} $
	13	$ 1 + 1.00e3iT - 4.82e6T^{2} $
	17	$ 1 + 3.71e3iT - 2.41e7T^{2} $
	19	$ 1 - 7.86e3iT - 4.70e7T^{2} $
	23	$ 1 - 1.95e4iT - 1.48e8T^{2} $
	29	$ 1 + 4.25e4T + 5.94e8T^{2} $
	31	$ 1 + 3.91e4T + 8.87e8T^{2} $
	37	$ 1 + 6.17e4iT - 2.56e9T^{2} $
	41	$ 1 - 1.33e4iT - 4.75e9T^{2} $
	43	$ 1 + 5.79e4iT - 6.32e9T^{2} $
	47	$ 1 - 5.19e4iT - 1.07e10T^{2} $
	53	$ 1 + 5.29e4T + 2.21e10T^{2} $
	59	$ 1 - 9.58e4T + 4.21e10T^{2} $
	61	$ 1 - 2.48e5iT - 5.15e10T^{2} $
	67	$ 1 - 1.70e5iT - 9.04e10T^{2} $
	71	$ 1 + 4.77e5iT - 1.28e11T^{2} $
	73	$ 1 + 4.81e5T + 1.51e11T^{2} $
	79	$ 1 - 3.58e5T + 2.43e11T^{2} $
	83	$ 1 + 8.58e4T + 3.26e11T^{2} $
	89	$ 1 + 1.09e6iT - 4.96e11T^{2} $
	97	$ 1 - 1.01e6T + 8.32e11T^{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

−9.739138057536825519153176542051, −9.093666505195063840745801848401, −7.64959717014358385778575872366, −7.35235404856391907822089833745, −5.82499486001850154182373378407, −5.50817701293702865388708338957, −3.80971715843418832190381841279, −3.16615828973889145191188294365, −1.96666632102925834949343283020, −0.24825221400291842748104446066, 0.29861401265616715383360129659, 2.11079515696651163026729153359, 3.05696674249060547498484266980, 4.04802435491322664076295039940, 5.27138047843585134620603073402, 6.24672452106796031980588167851, 7.01921504908585295914218359871, 8.006375623494811485078734586187, 9.028860655658179766586211748174, 9.790957092823906269299865484551

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(0.4139697591\)
\(L(\frac12)\)	\(\approx\)	\(0.4139697591\)
\(L(4)\)		not available
\(L(1)\)		not available

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