LMFDB - L-function 2-2736-12.11-c1-0-2

Dirichlet series

L(s) = 1

− 2.62i·5^-s − 0.815i·7^-s − 2.98·11^-s − 1.44·13^-s + 8.05i·17^-s + i·19^-s − 2.16·23^-s − 1.89·25^-s + 9.88i·29^-s − 10.1i·31^-s − 2.14·35^-s − 7.24·37^-s − 3.87i·41^-s + 11.2i·43^-s − 12.5·47^-s + ⋯

L(s) = 1

− 1.17i·5^-s − 0.308i·7^-s − 0.901·11^-s − 0.400·13^-s + 1.95i·17^-s + 0.229i·19^-s − 0.450·23^-s − 0.379·25^-s + 1.83i·29^-s − 1.83i·31^-s − 0.361·35^-s − 1.19·37^-s − 0.605i·41^-s + 1.71i·43^-s − 1.83·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.0917 - 0.995i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (2015, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ 0.0917 - 0.995i)$

Particular Values

$L(1)$	$\approx$	$0.7884958083$
$L(\frac12)$	$\approx$	$0.7884958083$
$L(\frac{3}{2})$		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 $
	19	$ 1 - iT $
good	5	$ 1 + 2.62iT - 5T^{2} $
	7	$ 1 + 0.815iT - 7T^{2} $
	11	$ 1 + 2.98T + 11T^{2} $
	13	$ 1 + 1.44T + 13T^{2} $
	17	$ 1 - 8.05iT - 17T^{2} $
	23	$ 1 + 2.16T + 23T^{2} $
	29	$ 1 - 9.88iT - 29T^{2} $
	31	$ 1 + 10.1iT - 31T^{2} $
	37	$ 1 + 7.24T + 37T^{2} $
	41	$ 1 + 3.87iT - 41T^{2} $
	43	$ 1 - 11.2iT - 43T^{2} $
	47	$ 1 + 12.5T + 47T^{2} $
	53	$ 1 - 8.04iT - 53T^{2} $
	59	$ 1 - 9.48T + 59T^{2} $
	61	$ 1 - 4.22T + 61T^{2} $
	67	$ 1 - 11.4iT - 67T^{2} $
	71	$ 1 - 4.11T + 71T^{2} $
	73	$ 1 - 12.1T + 73T^{2} $
	79	$ 1 + 15.4iT - 79T^{2} $
	83	$ 1 - 0.745T + 83T^{2} $
	89	$ 1 - 6.82iT - 89T^{2} $
	97	$ 1 + 0.975T + 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

−8.865989889878227217523146094847, −8.236176864154592466685124234035, −7.74264505653722573585963571747, −6.70356310416763678100991165830, −5.76322895207143158946358049136, −5.14126915951050505586498145473, −4.31344265669489960903230060895, −3.52316959362500964844541933599, −2.18069803510302803786674286543, −1.19850775451227425090400772429, 0.26034954479276456627346939989, 2.20386791277534685981027727240, 2.79014174830700435222777950247, 3.62413569228263051109330014543, 4.96983521650862776855713634569, 5.37944396906178265644668075613, 6.66306086072442920252270404839, 6.96663857204377782804854112755, 7.81953594875948451338002631458, 8.560179131588900568520066031814

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7884958083\)
\(L(\frac12)\)	\(\approx\)	\(0.7884958083\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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