LMFDB - L-function 2-5472-12.11-c1-0-32

Dirichlet series

L(s) = 1

− 0.896i·5^-s − 4.46i·7^-s − 2.96·11^-s + 5.46·13^-s + 7.20i·17^-s − i·19^-s + 0.378·23^-s + 4.19·25^-s + 8.48i·29^-s + 7.46i·31^-s − 4.00·35^-s + 9.46·37^-s − 2.82i·41^-s + 2.26i·43^-s + 8.24·47^-s + ⋯

L(s) = 1

− 0.400i·5^-s − 1.68i·7^-s − 0.894·11^-s + 1.51·13^-s + 1.74i·17^-s − 0.229i·19^-s + 0.0790·23^-s + 0.839·25^-s + 1.57i·29^-s + 1.34i·31^-s − 0.676·35^-s + 1.55·37^-s − 0.441i·41^-s + 0.345i·43^-s + 1.20·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5472$ = $2^{5} \cdot 3^{2} \cdot 19$
Sign:	$0.985 + 0.169i$
Analytic conductor:	$43.6941$
Root analytic conductor:	$6.61015$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5472} (2015, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5472,\ (\ :1/2),\ 0.985 + 0.169i)$

Particular Values

$L(1)$	$\approx$	$2.006641802$
$L(\frac12)$	$\approx$	$2.006641802$
$L(\frac{3}{2})$		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 + 0.896iT - 5T^{2} $
	7	$ 1 + 4.46iT - 7T^{2} $
	11	$ 1 + 2.96T + 11T^{2} $
	13	$ 1 - 5.46T + 13T^{2} $
	17	$ 1 - 7.20iT - 17T^{2} $
	23	$ 1 - 0.378T + 23T^{2} $
	29	$ 1 - 8.48iT - 29T^{2} $
	31	$ 1 - 7.46iT - 31T^{2} $
	37	$ 1 - 9.46T + 37T^{2} $
	41	$ 1 + 2.82iT - 41T^{2} $
	43	$ 1 - 2.26iT - 43T^{2} $
	47	$ 1 - 8.24T + 47T^{2} $
	53	$ 1 - 7.45iT - 53T^{2} $
	59	$ 1 + 3.10T + 59T^{2} $
	61	$ 1 + 7.19T + 61T^{2} $
	67	$ 1 - 4.92iT - 67T^{2} $
	71	$ 1 - 9.52T + 71T^{2} $
	73	$ 1 - 7.92T + 73T^{2} $
	79	$ 1 - 11.4iT - 79T^{2} $
	83	$ 1 - 8.10T + 83T^{2} $
	89	$ 1 - 5.93iT - 89T^{2} $
	97	$ 1 - 10.9T + 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

−8.226667414844473953460546629934, −7.44135938812025581287712483492, −6.75781209477520582941334079372, −6.06412728912751012345538338416, −5.18853424081665001450676125034, −4.34430214313415384523364714080, −3.77751508740939711624390340243, −2.99624588879715569989980724635, −1.49054974118274847998084371216, −0.928979000101356252700590317226, 0.68510350966381951889653496718, 2.28216750662206468359274768315, 2.63125654288602632839344457085, 3.54355567371487407755740311178, 4.66753138548580644082650051625, 5.38619654448860573874193055479, 6.06551354535882498466479113980, 6.50586856323444401345810221215, 7.77667502960119432259904808252, 7.989767274227990756186107319826

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

\(L(1)\)	\(\approx\)	\(2.006641802\)
\(L(\frac12)\)	\(\approx\)	\(2.006641802\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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