LMFDB - L-function 2-2312-17.16-c1-0-36

Dirichlet series

L(s) = 1

+ 0.0750i·3^-s − 2.06i·5^-s + 2.86i·7^-s + 2.99·9^-s − 1.65i·11^-s − 3.96·13^-s + 0.155·15^-s + 5.57·19^-s − 0.214·21^-s − 5.28i·23^-s + 0.717·25^-s + 0.449i·27^-s + 6.99i·29^-s + 6.39i·31^-s + 0.124·33^-s + ⋯

L(s) = 1

+ 0.0433i·3^-s − 0.925i·5^-s + 1.08i·7^-s + 0.998·9^-s − 0.498i·11^-s − 1.09·13^-s + 0.0401·15^-s + 1.27·19^-s − 0.0468·21^-s − 1.10i·23^-s + 0.143·25^-s + 0.0865i·27^-s + 1.29i·29^-s + 1.14i·31^-s + 0.0215·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2312$ = $2^{3} \cdot 17^{2}$
Sign:	$0.911 + 0.410i$
Analytic conductor:	$18.4614$
Root analytic conductor:	$4.29667$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2312} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2312,\ (\ :1/2),\ 0.911 + 0.410i)$

Particular Values

$L(1)$	$\approx$	$1.895143418$
$L(\frac12)$	$\approx$	$1.895143418$
$L(\frac{3}{2})$		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 $
	17	$ 1 $
good	3	$ 1 - 0.0750iT - 3T^{2} $
	5	$ 1 + 2.06iT - 5T^{2} $
	7	$ 1 - 2.86iT - 7T^{2} $
	11	$ 1 + 1.65iT - 11T^{2} $
	13	$ 1 + 3.96T + 13T^{2} $
	19	$ 1 - 5.57T + 19T^{2} $
	23	$ 1 + 5.28iT - 23T^{2} $
	29	$ 1 - 6.99iT - 29T^{2} $
	31	$ 1 - 6.39iT - 31T^{2} $
	37	$ 1 + 5.70iT - 37T^{2} $
	41	$ 1 + 9.87iT - 41T^{2} $
	43	$ 1 - 6.67T + 43T^{2} $
	47	$ 1 - 4.43T + 47T^{2} $
	53	$ 1 + 8.27T + 53T^{2} $
	59	$ 1 - 2.42T + 59T^{2} $
	61	$ 1 + 1.81iT - 61T^{2} $
	67	$ 1 - 15.8T + 67T^{2} $
	71	$ 1 + 8.54iT - 71T^{2} $
	73	$ 1 - 2.30iT - 73T^{2} $
	79	$ 1 - 14.0iT - 79T^{2} $
	83	$ 1 - 1.00T + 83T^{2} $
	89	$ 1 + 16.2T + 89T^{2} $
	97	$ 1 - 13.8iT - 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

−9.031951605813197146100144217236, −8.347770730675430363513534362389, −7.38343015206697057766501848563, −6.73818414306895739689684368196, −5.44260173570195303385051633610, −5.18370478745792616512380723977, −4.24483827827164255005432805622, −3.08254490731266233655483536250, −2.05090378343280438273026682030, −0.842532744329897086690209449791, 1.00738601406553312603587151721, 2.28324858023669101244623511687, 3.31241083573192986585822005889, 4.21619428132007225713304917283, 4.90961192677671730215715651037, 6.10159702169724484258248315605, 6.97541602754805961966536767333, 7.49828858045611539958699227143, 7.80306039689359788265651798783, 9.430960590599455479804769112648

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.895143418\)
\(L(\frac12)\)	\(\approx\)	\(1.895143418\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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