LMFDB - L-function 2-8280-69.68-c1-0-38

Dirichlet series

L(s) = 1

− 5^-s − 3.18i·7^-s − 3.95·11^-s − 1.35·13^-s − 5.36·17^-s + 6.29i·19^-s + (4.77 + 0.395i)23^-s + 25^-s + 1.07i·29^-s + 2.28·31^-s + 3.18i·35^-s + 6.80i·37^-s − 5.38i·41^-s + 13.0i·43^-s − 12.0i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s − 1.20i·7^-s − 1.19·11^-s − 0.377·13^-s − 1.30·17^-s + 1.44i·19^-s + (0.996 + 0.0825i)23^-s + 0.200·25^-s + 0.199i·29^-s + 0.410·31^-s + 0.538i·35^-s + 1.11i·37^-s − 0.841i·41^-s + 1.98i·43^-s − 1.76i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8280$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$0.766 + 0.642i$
Analytic conductor:	$66.1161$
Root analytic conductor:	$8.13118$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8280} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8280,\ (\ :1/2),\ 0.766 + 0.642i)$

Particular Values

$L(1)$	$\approx$	$1.127187165$
$L(\frac12)$	$\approx$	$1.127187165$
$L(\frac{3}{2})$		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 + T $
	23	$ 1 + (-4.77 - 0.395i)T $
good	7	$ 1 + 3.18iT - 7T^{2} $
	11	$ 1 + 3.95T + 11T^{2} $
	13	$ 1 + 1.35T + 13T^{2} $
	17	$ 1 + 5.36T + 17T^{2} $
	19	$ 1 - 6.29iT - 19T^{2} $
	29	$ 1 - 1.07iT - 29T^{2} $
	31	$ 1 - 2.28T + 31T^{2} $
	37	$ 1 - 6.80iT - 37T^{2} $
	41	$ 1 + 5.38iT - 41T^{2} $
	43	$ 1 - 13.0iT - 43T^{2} $
	47	$ 1 + 12.0iT - 47T^{2} $
	53	$ 1 + 0.663T + 53T^{2} $
	59	$ 1 - 8.88iT - 59T^{2} $
	61	$ 1 + 0.778iT - 61T^{2} $
	67	$ 1 - 8.99iT - 67T^{2} $
	71	$ 1 + 7.19iT - 71T^{2} $
	73	$ 1 - 9.84T + 73T^{2} $
	79	$ 1 + 2.44iT - 79T^{2} $
	83	$ 1 + 14.5T + 83T^{2} $
	89	$ 1 - 17.0T + 89T^{2} $
	97	$ 1 - 5.33iT - 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

−7.68271711309186659149452803881, −7.13156688394287448087437196796, −6.53093073502365989944170460092, −5.58564445214052750019118062664, −4.77717436716225657010779298529, −4.27523408405008316998000981550, −3.43190972510765335922946914523, −2.66900217189001747438453942651, −1.57946453784192433435124649360, −0.44548506200946710046567305462, 0.57917739645569342600192296297, 2.30576177733303189505381302109, 2.49629917013264883607146775228, 3.45320633727472535221354304196, 4.67856309670012988965954909717, 4.92777987817405690425757105978, 5.74898003279246189909835721242, 6.56412648487993835554951989464, 7.21161321514506300903546315839, 7.88536818561781710393721387227

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.127187165\)
\(L(\frac12)\)	\(\approx\)	\(1.127187165\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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