LMFDB - L-function 2-2016-3.2-c2-0-32

Dirichlet series

L(s) = 1

− 3.85i·5^-s + 2.64·7^-s − 6.89i·11^-s + 20.9·13^-s + 3.42i·17^-s + 11.1·19^-s + 25.1i·23^-s + 10.1·25^-s + 0.948i·29^-s + 6.63·31^-s − 10.2i·35^-s + 5.63·37^-s + 30.2i·41^-s − 16.6·43^-s − 9.40i·47^-s + ⋯

L(s) = 1

− 0.771i·5^-s + 0.377·7^-s − 0.626i·11^-s + 1.60·13^-s + 0.201i·17^-s + 0.586·19^-s + 1.09i·23^-s + 0.405·25^-s + 0.0326i·29^-s + 0.214·31^-s − 0.291i·35^-s + 0.152·37^-s + 0.737i·41^-s − 0.388·43^-s − 0.200i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2016$ = $2^{5} \cdot 3^{2} \cdot 7$
Sign:	$0.816 + 0.577i$
Analytic conductor:	$54.9320$
Root analytic conductor:	$7.41161$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2016} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2016,\ (\ :1),\ 0.816 + 0.577i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.477633844$
$L(\frac12)$	$\approx$	$2.477633844$
$L(2)$		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 - 2.64T $
good	5	$ 1 + 3.85iT - 25T^{2} $
	11	$ 1 + 6.89iT - 121T^{2} $
	13	$ 1 - 20.9T + 169T^{2} $
	17	$ 1 - 3.42iT - 289T^{2} $
	19	$ 1 - 11.1T + 361T^{2} $
	23	$ 1 - 25.1iT - 529T^{2} $
	29	$ 1 - 0.948iT - 841T^{2} $
	31	$ 1 - 6.63T + 961T^{2} $
	37	$ 1 - 5.63T + 1.36e3T^{2} $
	41	$ 1 - 30.2iT - 1.68e3T^{2} $
	43	$ 1 + 16.6T + 1.84e3T^{2} $
	47	$ 1 + 9.40iT - 2.20e3T^{2} $
	53	$ 1 - 44.4iT - 2.80e3T^{2} $
	59	$ 1 + 52.2iT - 3.48e3T^{2} $
	61	$ 1 - 48.5T + 3.72e3T^{2} $
	67	$ 1 - 63.5T + 4.48e3T^{2} $
	71	$ 1 + 35.6iT - 5.04e3T^{2} $
	73	$ 1 - 19.4T + 5.32e3T^{2} $
	79	$ 1 + 152.T + 6.24e3T^{2} $
	83	$ 1 - 97.7iT - 6.88e3T^{2} $
	89	$ 1 + 99.3iT - 7.92e3T^{2} $
	97	$ 1 - 133.T + 9.40e3T^{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

−8.717714070000217556060487719944, −8.328986636955637089468807423570, −7.46842007903165263286201361157, −6.38107145041483286632354298959, −5.67617272091581600418647193703, −4.91984402916746378876882051275, −3.91229000536644015467283733876, −3.13257893536481733644815582209, −1.60472375056197087017864861283, −0.852377659352913214677462607820, 0.951646259522528663943469192525, 2.15880836050267518540898463590, 3.17753853402518172565752088835, 4.05484148030891026458302199713, 5.00495147055178583361551787591, 5.99034075651962454918767438012, 6.71952582614408409061147492100, 7.38801152794118222641786314043, 8.348317467010147862833871625929, 8.897923241296343776620473416016

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.477633844\)
\(L(\frac12)\)	\(\approx\)	\(2.477633844\)
\(L(2)\)		not available
\(L(1)\)		not available

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