LMFDB - L-function 2-5328-37.36-c1-0-85

Dirichlet series

L(s) = 1

− 0.791i·5^-s + 2·7^-s − 0.791·11^-s − 3.79i·13^-s − 7.58i·17^-s − 1.58i·19^-s − 3.79i·23^-s + 4.37·25^-s − 3.79i·29^-s + 8.37i·31^-s − 1.58i·35^-s + (4 − 4.58i)37^-s − 9.79·41^-s + 6i·43^-s − 7.58·47^-s + ⋯

L(s) = 1

− 0.353i·5^-s + 0.755·7^-s − 0.238·11^-s − 1.05i·13^-s − 1.83i·17^-s − 0.363i·19^-s − 0.790i·23^-s + 0.874·25^-s − 0.704i·29^-s + 1.50i·31^-s − 0.267i·35^-s + (0.657 − 0.753i)37^-s − 1.52·41^-s + 0.914i·43^-s − 1.10·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5328$ = $2^{4} \cdot 3^{2} \cdot 37$
Sign:	$-0.657 + 0.753i$
Analytic conductor:	$42.5442$
Root analytic conductor:	$6.52259$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5328} (2737, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5328,\ (\ :1/2),\ -0.657 + 0.753i)$

Particular Values

$L(1)$	$\approx$	$1.487124221$
$L(\frac12)$	$\approx$	$1.487124221$
$L(\frac{3}{2})$		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 + (-4 + 4.58i)T $
good	5	$ 1 + 0.791iT - 5T^{2} $
	7	$ 1 - 2T + 7T^{2} $
	11	$ 1 + 0.791T + 11T^{2} $
	13	$ 1 + 3.79iT - 13T^{2} $
	17	$ 1 + 7.58iT - 17T^{2} $
	19	$ 1 + 1.58iT - 19T^{2} $
	23	$ 1 + 3.79iT - 23T^{2} $
	29	$ 1 + 3.79iT - 29T^{2} $
	31	$ 1 - 8.37iT - 31T^{2} $
	41	$ 1 + 9.79T + 41T^{2} $
	43	$ 1 - 6iT - 43T^{2} $
	47	$ 1 + 7.58T + 47T^{2} $
	53	$ 1 - 1.58T + 53T^{2} $
	59	$ 1 - 1.58iT - 59T^{2} $
	61	$ 1 - 12.7iT - 61T^{2} $
	67	$ 1 + 6.37T + 67T^{2} $
	71	$ 1 + 9.16T + 71T^{2} $
	73	$ 1 + 4.37T + 73T^{2} $
	79	$ 1 + 8.20iT - 79T^{2} $
	83	$ 1 - 15.1T + 83T^{2} $
	89	$ 1 + 6iT - 89T^{2} $
	97	$ 1 + 13.5iT - 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

−7.922074394080018330593657089853, −7.31404380415847478864196715316, −6.57251479840028539490529129552, −5.57731323029372665349611806895, −4.88626767738481861295572905119, −4.57690794041386772855310075092, −3.16981029387750038769469854357, −2.65271557933442384525385034588, −1.38120577993471827512257446437, −0.38866470837453453994129024922, 1.46716471532493974656897368789, 2.02126707516529472084149889084, 3.25946107509411900262404181450, 3.99357556739637539366079289816, 4.77328374351445458038114869161, 5.55365449580979094917646876040, 6.41249616291227043269360239040, 6.90253224922564556975669682568, 7.982054992979609082638881304523, 8.196805953379555162271466391843

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.487124221\)
\(L(\frac12)\)	\(\approx\)	\(1.487124221\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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