LMFDB - L-function 2-2016-12.11-c1-0-10

Dirichlet series

L(s) = 1

− i·7^-s + 4.24·11^-s − 2·13^-s + 2.82i·17^-s + 4i·19^-s − 1.41·23^-s + 5·25^-s − 7.07i·29^-s + 8i·31^-s + 8·37^-s − 5.65i·41^-s + 4i·43^-s + 2.82·47^-s − 49^-s − 7.07i·53^-s + ⋯

L(s) = 1

− 0.377i·7^-s + 1.27·11^-s − 0.554·13^-s + 0.685i·17^-s + 0.917i·19^-s − 0.294·23^-s + 25^-s − 1.31i·29^-s + 1.43i·31^-s + 1.31·37^-s − 0.883i·41^-s + 0.609i·43^-s + 0.412·47^-s − 0.142·49^-s − 0.971i·53^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$2016$ = $2^{5} \cdot 3^{2} \cdot 7$
Sign:	$0.985 - 0.169i$
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:	$(2,\ 2016,\ (\ :1/2),\ 0.985 - 0.169i)$

Particular Values

$L(1)$	$\approx$	$1.853248592$
$L(\frac12)$	$\approx$	$1.853248592$
$L(\frac{3}{2})$		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 - 5T^{2} $
	11	$ 1 - 4.24T + 11T^{2} $
	13	$ 1 + 2T + 13T^{2} $
	17	$ 1 - 2.82iT - 17T^{2} $
	19	$ 1 - 4iT - 19T^{2} $
	23	$ 1 + 1.41T + 23T^{2} $
	29	$ 1 + 7.07iT - 29T^{2} $
	31	$ 1 - 8iT - 31T^{2} $
	37	$ 1 - 8T + 37T^{2} $
	41	$ 1 + 5.65iT - 41T^{2} $
	43	$ 1 - 4iT - 43T^{2} $
	47	$ 1 - 2.82T + 47T^{2} $
	53	$ 1 + 7.07iT - 53T^{2} $
	59	$ 1 + 2.82T + 59T^{2} $
	61	$ 1 + 2T + 61T^{2} $
	67	$ 1 + 2iT - 67T^{2} $
	71	$ 1 - 12.7T + 71T^{2} $
	73	$ 1 - 14T + 73T^{2} $
	79	$ 1 - 2iT - 79T^{2} $
	83	$ 1 - 5.65T + 83T^{2} $
	89	$ 1 - 11.3iT - 89T^{2} $
	97	$ 1 - 14T + 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

−9.239347057594772541902740436023, −8.350914141440601095309266656524, −7.67260951977598307797949077547, −6.68891026375069002630612395605, −6.19939681344748387694021460643, −5.09267202564017395923981964273, −4.15868948330928006981508966118, −3.48894967884380448386831014864, −2.16240653847088611103750905835, −1.01201643720519979063808924921, 0.875690472027791459703225736434, 2.25325465044827825658821122845, 3.18669320560140747191374876143, 4.31445478422082908968124177684, 5.01420008933513424535478159925, 6.04284491757083467880792530772, 6.78922865009807992162703369787, 7.47227152870880765035683676650, 8.459518982982464605441195068002, 9.339483418589480735975539892339

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.853248592\)
\(L(\frac12)\)	\(\approx\)	\(1.853248592\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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