LMFDB - L-function 2-8280-69.68-c1-0-21

Dirichlet series

L(s) = 1

− 5^-s + 2.84i·7^-s − 2.95·11^-s − 0.451·13^-s + 4.38·17^-s − 7.84i·19^-s + (3.10 + 3.65i)23^-s + 25^-s + 3.01i·29^-s − 6.09·31^-s − 2.84i·35^-s − 0.619i·37^-s − 5.35i·41^-s + 3.75i·43^-s + 5.23i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s + 1.07i·7^-s − 0.891·11^-s − 0.125·13^-s + 1.06·17^-s − 1.80i·19^-s + (0.647 + 0.762i)23^-s + 0.200·25^-s + 0.560i·29^-s − 1.09·31^-s − 0.481i·35^-s − 0.101i·37^-s − 0.836i·41^-s + 0.572i·43^-s + 0.763i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0887 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0887 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8280$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.0887 - 0.996i$
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:	$(2,\ 8280,\ (\ :1/2),\ -0.0887 - 0.996i)$

Particular Values

$L(1)$	$\approx$	$1.276209431$
$L(\frac12)$	$\approx$	$1.276209431$
$L(\frac{3}{2})$		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 + (-3.10 - 3.65i)T $
good	7	$ 1 - 2.84iT - 7T^{2} $
	11	$ 1 + 2.95T + 11T^{2} $
	13	$ 1 + 0.451T + 13T^{2} $
	17	$ 1 - 4.38T + 17T^{2} $
	19	$ 1 + 7.84iT - 19T^{2} $
	29	$ 1 - 3.01iT - 29T^{2} $
	31	$ 1 + 6.09T + 31T^{2} $
	37	$ 1 + 0.619iT - 37T^{2} $
	41	$ 1 + 5.35iT - 41T^{2} $
	43	$ 1 - 3.75iT - 43T^{2} $
	47	$ 1 - 5.23iT - 47T^{2} $
	53	$ 1 - 14.0T + 53T^{2} $
	59	$ 1 + 11.8iT - 59T^{2} $
	61	$ 1 - 9.81iT - 61T^{2} $
	67	$ 1 - 12.3iT - 67T^{2} $
	71	$ 1 + 13.6iT - 71T^{2} $
	73	$ 1 - 1.07T + 73T^{2} $
	79	$ 1 - 8.71iT - 79T^{2} $
	83	$ 1 - 3.20T + 83T^{2} $
	89	$ 1 + 3.64T + 89T^{2} $
	97	$ 1 - 8.46iT - 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

−7.909155653289915160154184072473, −7.34129016978650373726845660698, −6.76907194302992188031854428631, −5.57150870045104864175744606791, −5.40902982809934627267988774189, −4.61750883030199761970000008248, −3.55066300840965922814637877244, −2.85786185624484312052946918044, −2.20310786823272703208662816196, −0.895570293787353494986069769893, 0.37304532714953570124214991847, 1.36686570914359103901334244816, 2.47958464088409696827767342096, 3.51119956483186872410287252511, 3.89715098959799497826825203335, 4.81419564638070618652969153875, 5.52151880804564893420129250235, 6.21694099725815739835632897033, 7.26332163153325562735160602613, 7.49476482794833337003958810441

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.276209431\)
\(L(\frac12)\)	\(\approx\)	\(1.276209431\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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