LMFDB - L-function 2-1152-8.5-c3-0-41

Dirichlet series

L(s) = 1

− 17.4i·5^-s − 2.99·7^-s + 10.6i·11^-s + 43.3i·13^-s + 37.8·17^-s − 79.8i·19^-s + 191.·23^-s − 178.·25^-s + 138. i·29^-s + 212.·31^-s + 52.1i·35^-s − 270. i·37^-s + 441.·41^-s + 64.1i·43^-s + 436.·47^-s + ⋯

L(s) = 1

− 1.55i·5^-s − 0.161·7^-s + 0.291i·11^-s + 0.924i·13^-s + 0.540·17^-s − 0.964i·19^-s + 1.73·23^-s − 1.43·25^-s + 0.889i·29^-s + 1.22·31^-s + 0.251i·35^-s − 1.20i·37^-s + 1.68·41^-s + 0.227i·43^-s + 1.35·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$i$
Analytic conductor:	$67.9702$
Root analytic conductor:	$8.24440$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :3/2),\ i)$

Particular Values

$L(2)$	$\approx$	$2.024518303$
$L(\frac12)$	$\approx$	$2.024518303$
$L(\frac{5}{2})$		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 + 17.4iT - 125T^{2} $
	7	$ 1 + 2.99T + 343T^{2} $
	11	$ 1 - 10.6iT - 1.33e3T^{2} $
	13	$ 1 - 43.3iT - 2.19e3T^{2} $
	17	$ 1 - 37.8T + 4.91e3T^{2} $
	19	$ 1 + 79.8iT - 6.85e3T^{2} $
	23	$ 1 - 191.T + 1.21e4T^{2} $
	29	$ 1 - 138. iT - 2.43e4T^{2} $
	31	$ 1 - 212.T + 2.97e4T^{2} $
	37	$ 1 + 270. iT - 5.06e4T^{2} $
	41	$ 1 - 441.T + 6.89e4T^{2} $
	43	$ 1 - 64.1iT - 7.95e4T^{2} $
	47	$ 1 - 436.T + 1.03e5T^{2} $
	53	$ 1 + 278. iT - 1.48e5T^{2} $
	59	$ 1 + 830. iT - 2.05e5T^{2} $
	61	$ 1 - 724. iT - 2.26e5T^{2} $
	67	$ 1 + 859. iT - 3.00e5T^{2} $
	71	$ 1 + 681.T + 3.57e5T^{2} $
	73	$ 1 + 785.T + 3.89e5T^{2} $
	79	$ 1 + 1.01e3T + 4.93e5T^{2} $
	83	$ 1 - 467. iT - 5.71e5T^{2} $
	89	$ 1 + 510.T + 7.04e5T^{2} $
	97	$ 1 + 234.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

−9.082317791964432618961436192461, −8.696897889485989905644804789777, −7.58020778845096596192826576124, −6.78865748945506162008002634882, −5.64391547531652107107325396367, −4.82315385099333061410254280144, −4.24454605072518016685984935119, −2.85243219494961954132908540998, −1.49508914497466477179276019444, −0.59200752991177752465189296636, 1.03831991580905578803413510145, 2.74695332104695814375032805201, 3.09723682831516419102028626658, 4.28852507159995542693777041533, 5.66678552318488415548793401747, 6.23351540323450038393346052382, 7.21511579718168299776845782335, 7.78261672797151904285233618058, 8.768573805642631131465774585551, 9.965612169736428387958457661649

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.024518303\)
\(L(\frac12)\)	\(\approx\)	\(2.024518303\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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