LMFDB - L-function 2-5328-12.11-c1-0-26

Dirichlet series

L(s) = 1

+ 0.617i·5^-s + 2.19i·7^-s + 0.928·11^-s − 0.951·13^-s + 0.493i·17^-s − 2.39i·19^-s + 5.17·23^-s + 4.61·25^-s + 4.22i·29^-s − 2.47i·31^-s − 1.35·35^-s − 37^-s − 7.06i·41^-s + 9.01i·43^-s − 0.0135·47^-s + ⋯

L(s) = 1

+ 0.276i·5^-s + 0.827i·7^-s + 0.279·11^-s − 0.263·13^-s + 0.119i·17^-s − 0.548i·19^-s + 1.08·23^-s + 0.923·25^-s + 0.784i·29^-s − 0.445i·31^-s − 0.228·35^-s − 0.164·37^-s − 1.10i·41^-s + 1.37i·43^-s − 0.00197·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5328$ = $2^{4} \cdot 3^{2} \cdot 37$
Sign:	$0.418 - 0.908i$
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:	$(2,\ 5328,\ (\ :1/2),\ 0.418 - 0.908i)$

Particular Values

$L(1)$	$\approx$	$1.904127102$
$L(\frac12)$	$\approx$	$1.904127102$
$L(\frac{3}{2})$		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 - 0.617iT - 5T^{2} $
	7	$ 1 - 2.19iT - 7T^{2} $
	11	$ 1 - 0.928T + 11T^{2} $
	13	$ 1 + 0.951T + 13T^{2} $
	17	$ 1 - 0.493iT - 17T^{2} $
	19	$ 1 + 2.39iT - 19T^{2} $
	23	$ 1 - 5.17T + 23T^{2} $
	29	$ 1 - 4.22iT - 29T^{2} $
	31	$ 1 + 2.47iT - 31T^{2} $
	41	$ 1 + 7.06iT - 41T^{2} $
	43	$ 1 - 9.01iT - 43T^{2} $
	47	$ 1 + 0.0135T + 47T^{2} $
	53	$ 1 + 7.85iT - 53T^{2} $
	59	$ 1 - 10.4T + 59T^{2} $
	61	$ 1 + 5.10T + 61T^{2} $
	67	$ 1 - 11.1iT - 67T^{2} $
	71	$ 1 + 0.138T + 71T^{2} $
	73	$ 1 + 6.72T + 73T^{2} $
	79	$ 1 + 10.7iT - 79T^{2} $
	83	$ 1 - 14.6T + 83T^{2} $
	89	$ 1 - 13.8iT - 89T^{2} $
	97	$ 1 - 9.48T + 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

−8.467022312106352113332166461425, −7.50230755899095049985689601509, −6.87446562144977991466944471644, −6.24582569453001763219674879794, −5.31770498543729065773810455596, −4.84058652276645155374450433476, −3.75853141166722311842933848481, −2.90726385129144246863581170293, −2.21821161123568988748825246336, −0.982266395245561976841077783435, 0.61484713994287356309786258433, 1.54308411945616707600407639192, 2.74069967225383829719789495307, 3.59935820605903130795056303695, 4.40676917261732121045717697133, 5.03723347972209623630274512736, 5.90631486729036116163057988994, 6.77627103444117834285988940548, 7.27564045406986038296335057639, 8.022061928233082616390091476074

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

\(L(1)\)	\(\approx\)	\(1.904127102\)
\(L(\frac12)\)	\(\approx\)	\(1.904127102\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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