LMFDB - L-function 2-1152-8.5-c1-0-10

Dirichlet series

L(s) = 1

− 4·7^-s + 4i·11^-s − 4i·13^-s + 2·17^-s − 4i·19^-s + 8·23^-s + 5·25^-s − 8i·29^-s + 4·31^-s − 4i·37^-s + 6·41^-s − 4i·43^-s − 8·47^-s + 9·49^-s + 8i·53^-s + ⋯

L(s) = 1

− 1.51·7^-s + 1.20i·11^-s − 1.10i·13^-s + 0.485·17^-s − 0.917i·19^-s + 1.66·23^-s + 25^-s − 1.48i·29^-s + 0.718·31^-s − 0.657i·37^-s + 0.937·41^-s − 0.609i·43^-s − 1.16·47^-s + 1.28·49^-s + 1.09i·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$9.19876$
Root analytic conductor:	$3.03294$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.260261735$
$L(\frac12)$	$\approx$	$1.260261735$
$L(\frac{3}{2})$		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 - 5T^{2} $
	7	$ 1 + 4T + 7T^{2} $
	11	$ 1 - 4iT - 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 - 2T + 17T^{2} $
	19	$ 1 + 4iT - 19T^{2} $
	23	$ 1 - 8T + 23T^{2} $
	29	$ 1 + 8iT - 29T^{2} $
	31	$ 1 - 4T + 31T^{2} $
	37	$ 1 + 4iT - 37T^{2} $
	41	$ 1 - 6T + 41T^{2} $
	43	$ 1 + 4iT - 43T^{2} $
	47	$ 1 + 8T + 47T^{2} $
	53	$ 1 - 8iT - 53T^{2} $
	59	$ 1 + 12iT - 59T^{2} $
	61	$ 1 + 12iT - 61T^{2} $
	67	$ 1 - 12iT - 67T^{2} $
	71	$ 1 + 8T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 - 4T + 79T^{2} $
	83	$ 1 - 4iT - 83T^{2} $
	89	$ 1 + 6T + 89T^{2} $
	97	$ 1 + 2T + 97T^{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

−9.663811347489229773588930741192, −9.132048518469446966080876486697, −7.973267919742689425604578360882, −7.08041437063497275499251406500, −6.50225767569213131207571378596, −5.44173536080275077641181238376, −4.52072352696192125743727723929, −3.27450664674083679853318110082, −2.57591982122599698652948702223, −0.66174165291503170838723340308, 1.12088616122945313007546739541, 2.96515159549670579511462281210, 3.43490745113121803273750169249, 4.73336276285199691977408898635, 5.85632511485663524756946117181, 6.55201621195319094357741740463, 7.21260709704559913010211181603, 8.502765385037461502213041733583, 9.062541882330379183500954998836, 9.836215025557699865033652175394

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.260261735\)
\(L(\frac12)\)	\(\approx\)	\(1.260261735\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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