LMFDB - L-function 2-5328-12.11-c1-0-67

Dirichlet series

L(s) = 1

− 3.05i·5^-s − 1.35i·7^-s − 5.12·11^-s + 5.32·13^-s − 4.54i·17^-s + 1.11i·19^-s − 0.226·23^-s − 4.34·25^-s − 3.49i·29^-s − 0.0119i·31^-s − 4.14·35^-s − 37^-s − 5.56i·41^-s − 6.47i·43^-s − 6.43·47^-s + ⋯

L(s) = 1

− 1.36i·5^-s − 0.512i·7^-s − 1.54·11^-s + 1.47·13^-s − 1.10i·17^-s + 0.254i·19^-s − 0.0471·23^-s − 0.869·25^-s − 0.648i·29^-s − 0.00213i·31^-s − 0.700·35^-s − 0.164·37^-s − 0.869i·41^-s − 0.987i·43^-s − 0.938·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5328$ = $2^{4} \cdot 3^{2} \cdot 37$
Sign:	$-0.995 - 0.0917i$
Analytic conductor:	$42.5442$
Root analytic conductor:	$6.52259$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5328} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5328,\ (\ :1/2),\ -0.995 - 0.0917i)$

Particular Values

$L(1)$	$\approx$	$1.060919536$
$L(\frac12)$	$\approx$	$1.060919536$
$L(\frac{3}{2})$		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 $
	37	$ 1 + T $
good	5	$ 1 + 3.05iT - 5T^{2} $
	7	$ 1 + 1.35iT - 7T^{2} $
	11	$ 1 + 5.12T + 11T^{2} $
	13	$ 1 - 5.32T + 13T^{2} $
	17	$ 1 + 4.54iT - 17T^{2} $
	19	$ 1 - 1.11iT - 19T^{2} $
	23	$ 1 + 0.226T + 23T^{2} $
	29	$ 1 + 3.49iT - 29T^{2} $
	31	$ 1 + 0.0119iT - 31T^{2} $
	41	$ 1 + 5.56iT - 41T^{2} $
	43	$ 1 + 6.47iT - 43T^{2} $
	47	$ 1 + 6.43T + 47T^{2} $
	53	$ 1 - 4.20iT - 53T^{2} $
	59	$ 1 + 4.13T + 59T^{2} $
	61	$ 1 + 0.618T + 61T^{2} $
	67	$ 1 + 10.2iT - 67T^{2} $
	71	$ 1 + 4.84T + 71T^{2} $
	73	$ 1 - 1.62T + 73T^{2} $
	79	$ 1 + 3.25iT - 79T^{2} $
	83	$ 1 - 12.9T + 83T^{2} $
	89	$ 1 - 11.4iT - 89T^{2} $
	97	$ 1 + 10.7T + 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

−7.922571299111170261620443680540, −7.29615355415017016707895006403, −6.26573916399303219366781836002, −5.46363817519542398341620380670, −4.98852821974584014772575071973, −4.18994912065796373553550054370, −3.36958606630819213873662681208, −2.27022644184990367333749599922, −1.15765027176195907488070708258, −0.29364745442485287466695343606, 1.52354157102479327043524541480, 2.59321152093798013419804373004, 3.13483572610161280783394354898, 3.93180877241895177337462611097, 5.03691748377801945070274282014, 5.85308703972917083948698306888, 6.34685961616927096768618389409, 7.05745671297242265418914087501, 7.964966775691913381916661609613, 8.330683475319353821774534487116

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

\(L(1)\)	\(\approx\)	\(1.060919536\)
\(L(\frac12)\)	\(\approx\)	\(1.060919536\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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