LMFDB - L-function 2-7920-33.32-c1-0-64

Dirichlet series

L(s) = 1

− i·5^-s + 0.112i·7^-s + (−2.27 + 2.41i)11^-s − 2.16i·13^-s − 3.32·17^-s + 2.43i·19^-s − 4.21i·23^-s − 25^-s + 9.11·29^-s + 6.54·31^-s + 0.112·35^-s − 3.38·37^-s − 8.32·41^-s + 6.72i·43^-s + 12.7i·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s + 0.0424i·7^-s + (−0.685 + 0.727i)11^-s − 0.599i·13^-s − 0.807·17^-s + 0.558i·19^-s − 0.878i·23^-s − 0.200·25^-s + 1.69·29^-s + 1.17·31^-s + 0.0189·35^-s − 0.556·37^-s − 1.29·41^-s + 1.02i·43^-s + 1.86i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7920$ = $2^{4} \cdot 3^{2} \cdot 5 \cdot 11$
Sign:	$-0.139 + 0.990i$
Analytic conductor:	$63.2415$
Root analytic conductor:	$7.95245$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7920} (3761, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7920,\ (\ :1/2),\ -0.139 + 0.990i)$

Particular Values

$L(1)$	$\approx$	$1.212176702$
$L(\frac12)$	$\approx$	$1.212176702$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
	11	$ 1 + (2.27 - 2.41i)T $
good	7	$ 1 - 0.112iT - 7T^{2} $
	13	$ 1 + 2.16iT - 13T^{2} $
	17	$ 1 + 3.32T + 17T^{2} $
	19	$ 1 - 2.43iT - 19T^{2} $
	23	$ 1 + 4.21iT - 23T^{2} $
	29	$ 1 - 9.11T + 29T^{2} $
	31	$ 1 - 6.54T + 31T^{2} $
	37	$ 1 + 3.38T + 37T^{2} $
	41	$ 1 + 8.32T + 41T^{2} $
	43	$ 1 - 6.72iT - 43T^{2} $
	47	$ 1 - 12.7iT - 47T^{2} $
	53	$ 1 + 10.6iT - 53T^{2} $
	59	$ 1 - 4.79iT - 59T^{2} $
	61	$ 1 + 13.8iT - 61T^{2} $
	67	$ 1 - 1.08T + 67T^{2} $
	71	$ 1 + 11.6iT - 71T^{2} $
	73	$ 1 - 3.88iT - 73T^{2} $
	79	$ 1 - 0.241iT - 79T^{2} $
	83	$ 1 + 0.512T + 83T^{2} $
	89	$ 1 + 13.9iT - 89T^{2} $
	97	$ 1 - 1.91T + 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

−7.994600174629126127335542069264, −6.80693635666716049904329775608, −6.41660341656566141274924879357, −5.46347014632318880027695606823, −4.73488766107110467823563427755, −4.36192651686744974167000382756, −3.15469265062911557066290844218, −2.48893641679594827324589683872, −1.49987295268174165427682271974, −0.32705892810783647094030164231, 0.946161073561950854143471628776, 2.20547038940533064132109792718, 2.84193587419542583868936222218, 3.69212860913143652202001341000, 4.50834164694517369967870062249, 5.26508313662544280669084963769, 5.96376693138367616413793629294, 6.90704491320063550232242131570, 7.02426185015613915566318292247, 8.201158518127824011608901871262

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.212176702\)
\(L(\frac12)\)	\(\approx\)	\(1.212176702\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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