LMFDB - L-function 2-2016-12.11-c1-0-17

Dirichlet series

L(s) = 1

− 2.82i·5^-s − i·7^-s + 1.41·11^-s + 6·13^-s + 5.65i·17^-s − 8i·19^-s + 1.41·23^-s − 3.00·25^-s + 4.24i·29^-s − 4i·31^-s − 2.82·35^-s + 8·37^-s − 8.48i·41^-s + 12i·43^-s + 2.82·47^-s + ⋯

L(s) = 1

− 1.26i·5^-s − 0.377i·7^-s + 0.426·11^-s + 1.66·13^-s + 1.37i·17^-s − 1.83i·19^-s + 0.294·23^-s − 0.600·25^-s + 0.787i·29^-s − 0.718i·31^-s − 0.478·35^-s + 1.31·37^-s − 1.32i·41^-s + 1.82i·43^-s + 0.412·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2016$ = $2^{5} \cdot 3^{2} \cdot 7$
Sign:	$0.169 + 0.985i$
Analytic conductor:	$16.0978$
Root analytic conductor:	$4.01221$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2016} (575, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2016,\ (\ :1/2),\ 0.169 + 0.985i)$

Particular Values

$L(1)$	$\approx$	$1.913883846$
$L(\frac12)$	$\approx$	$1.913883846$
$L(\frac{3}{2})$		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 $
	7	$ 1 + iT $
good	5	$ 1 + 2.82iT - 5T^{2} $
	11	$ 1 - 1.41T + 11T^{2} $
	13	$ 1 - 6T + 13T^{2} $
	17	$ 1 - 5.65iT - 17T^{2} $
	19	$ 1 + 8iT - 19T^{2} $
	23	$ 1 - 1.41T + 23T^{2} $
	29	$ 1 - 4.24iT - 29T^{2} $
	31	$ 1 + 4iT - 31T^{2} $
	37	$ 1 - 8T + 37T^{2} $
	41	$ 1 + 8.48iT - 41T^{2} $
	43	$ 1 - 12iT - 43T^{2} $
	47	$ 1 - 2.82T + 47T^{2} $
	53	$ 1 + 1.41iT - 53T^{2} $
	59	$ 1 + 14.1T + 59T^{2} $
	61	$ 1 - 2T + 61T^{2} $
	67	$ 1 + 14iT - 67T^{2} $
	71	$ 1 + 12.7T + 71T^{2} $
	73	$ 1 - 2T + 73T^{2} $
	79	$ 1 + 10iT - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 + 14.1iT - 89T^{2} $
	97	$ 1 + 6T + 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

−8.959781576887878776520882286849, −8.387680451886898598947776071903, −7.54564954217669064115901574526, −6.44061218308063451695588553613, −5.88712477898299920694646294413, −4.74583591199790861561420840714, −4.21013493065346535161220347645, −3.20985189463747322517125435449, −1.63789403645100692738610325000, −0.796277695346660624491462692128, 1.32360169918675921311023180693, 2.63647230669998500667089564230, 3.41191508142841726603185793721, 4.23133469647246136472888468125, 5.60541958274214311231796109574, 6.18444985092489853450887768189, 6.88986803356629324725339282748, 7.73696646361898686626810038290, 8.517655240402179822711426459265, 9.357924859795939649724571304025

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.913883846\)
\(L(\frac12)\)	\(\approx\)	\(1.913883846\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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