LMFDB - L-function 2-72e2-12.11-c1-0-19

Dirichlet series

L(s) = 1

− 0.517i·5^-s + 2.73i·7^-s − 0.378·11^-s + 2.46·13^-s + 3.34i·17^-s + 4.73i·19^-s + 1.41·23^-s + 4.73·25^-s − 6.83i·29^-s + 0.535i·31^-s + 1.41·35^-s + 4.26·37^-s + 5.27i·41^-s − 5.26i·43^-s − 9.52·47^-s + ⋯

L(s) = 1

− 0.231i·5^-s + 1.03i·7^-s − 0.114·11^-s + 0.683·13^-s + 0.811i·17^-s + 1.08i·19^-s + 0.294·23^-s + 0.946·25^-s − 1.26i·29^-s + 0.0962i·31^-s + 0.239·35^-s + 0.701·37^-s + 0.824i·41^-s − 0.803i·43^-s − 1.38·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$-i$
Analytic conductor:	$41.3944$
Root analytic conductor:	$6.43385$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5184} (5183, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5184,\ (\ :1/2),\ -i)$

Particular Values

$L(1)$	$\approx$	$1.721298134$
$L(\frac12)$	$\approx$	$1.721298134$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 0.517iT - 5T^{2} $
	7	$ 1 - 2.73iT - 7T^{2} $
	11	$ 1 + 0.378T + 11T^{2} $
	13	$ 1 - 2.46T + 13T^{2} $
	17	$ 1 - 3.34iT - 17T^{2} $
	19	$ 1 - 4.73iT - 19T^{2} $
	23	$ 1 - 1.41T + 23T^{2} $
	29	$ 1 + 6.83iT - 29T^{2} $
	31	$ 1 - 0.535iT - 31T^{2} $
	37	$ 1 - 4.26T + 37T^{2} $
	41	$ 1 - 5.27iT - 41T^{2} $
	43	$ 1 + 5.26iT - 43T^{2} $
	47	$ 1 + 9.52T + 47T^{2} $
	53	$ 1 + 2.44iT - 53T^{2} $
	59	$ 1 - 13.6T + 59T^{2} $
	61	$ 1 + 9.19T + 61T^{2} $
	67	$ 1 - 11.1iT - 67T^{2} $
	71	$ 1 + 16.1T + 71T^{2} $
	73	$ 1 + 10.2T + 73T^{2} $
	79	$ 1 - 14.5iT - 79T^{2} $
	83	$ 1 + 0.757T + 83T^{2} $
	89	$ 1 - 2.20iT - 89T^{2} $
	97	$ 1 - 14.3T + 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

−8.476531555501464119163139190805, −7.87354369884434437806121428847, −6.88466603528786761324750910023, −6.00187535031332701810872457045, −5.72254922153434532655491273009, −4.73442739724787383696342552417, −3.92804106137454020344468427924, −3.03969115847942410893393363585, −2.12867067442407274136352927433, −1.16146326582422151452989718248, 0.50453522316932575411769310788, 1.48096135660787156595538028525, 2.82687727854987411813108799857, 3.37840640875989391864278193881, 4.44904763984828985877539957146, 4.92760150546784516080962860868, 5.94292220671857157281657015876, 6.82160684525629642807029290119, 7.17497133585667547131774153035, 7.915167219744368275876947229077

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.721298134\)
\(L(\frac12)\)	\(\approx\)	\(1.721298134\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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