LMFDB - L-function 2-8280-69.68-c1-0-82

Dirichlet series

L(s) = 1

− 5^-s − 3.11i·7^-s + 4.47·11^-s + 3.14·13^-s + 0.883·17^-s − 5.10i·19^-s + (2.92 − 3.80i)23^-s + 25^-s − 6.05i·29^-s + 4.03·31^-s + 3.11i·35^-s − 1.93i·37^-s − 7.43i·41^-s − 7.84i·43^-s + 10.5i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s − 1.17i·7^-s + 1.35·11^-s + 0.873·13^-s + 0.214·17^-s − 1.17i·19^-s + (0.609 − 0.792i)23^-s + 0.200·25^-s − 1.12i·29^-s + 0.724·31^-s + 0.526i·35^-s − 0.317i·37^-s − 1.16i·41^-s − 1.19i·43^-s + 1.53i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0403 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0403 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8280$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.0403 + 0.999i$
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:	$(2,\ 8280,\ (\ :1/2),\ -0.0403 + 0.999i)$

Particular Values

$L(1)$	$\approx$	$2.176909973$
$L(\frac12)$	$\approx$	$2.176909973$
$L(\frac{3}{2})$		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 + (-2.92 + 3.80i)T $
good	7	$ 1 + 3.11iT - 7T^{2} $
	11	$ 1 - 4.47T + 11T^{2} $
	13	$ 1 - 3.14T + 13T^{2} $
	17	$ 1 - 0.883T + 17T^{2} $
	19	$ 1 + 5.10iT - 19T^{2} $
	29	$ 1 + 6.05iT - 29T^{2} $
	31	$ 1 - 4.03T + 31T^{2} $
	37	$ 1 + 1.93iT - 37T^{2} $
	41	$ 1 + 7.43iT - 41T^{2} $
	43	$ 1 + 7.84iT - 43T^{2} $
	47	$ 1 - 10.5iT - 47T^{2} $
	53	$ 1 + 0.0758T + 53T^{2} $
	59	$ 1 - 6.25iT - 59T^{2} $
	61	$ 1 + 10.8iT - 61T^{2} $
	67	$ 1 - 16.2iT - 67T^{2} $
	71	$ 1 - 12.8iT - 71T^{2} $
	73	$ 1 + 7.24T + 73T^{2} $
	79	$ 1 - 3.66iT - 79T^{2} $
	83	$ 1 + 8.57T + 83T^{2} $
	89	$ 1 - 10.7T + 89T^{2} $
	97	$ 1 - 7.27iT - 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

−7.45310597005428984653318354216, −6.97950811627186415507444464819, −6.44086151999351379599414451283, −5.62522959546014385196206124821, −4.47313315215659523309277268094, −4.16904269634792045198140104133, −3.48244837937178002958702299259, −2.50725247583490222220825438680, −1.20142808825903445148771451647, −0.61604535577140949120534999988, 1.16625969770455781288028098421, 1.82504505127986661918555559410, 3.15328130315819420704510056351, 3.50616310929317653308052898259, 4.46363726300280630918797297083, 5.22870911101391394343115069341, 6.09060353475913863636709466836, 6.39474813546071101323089575008, 7.32003400609517258956790005716, 8.099051734239449069438983921660

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.176909973\)
\(L(\frac12)\)	\(\approx\)	\(2.176909973\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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