LMFDB - L-function 2-2160-12.11-c1-0-21

Dirichlet series

L(s) = 1

− i·5^-s + 1.26i·7^-s − 1.26·11^-s − 4.19·13^-s + 5.19i·17^-s − 6.46i·19^-s + 7.73·23^-s − 25^-s − 2.19i·29^-s − 6.46i·31^-s + 1.26·35^-s + 2·37^-s − 8.19i·41^-s − 11.6i·43^-s − 2.53·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s + 0.479i·7^-s − 0.382·11^-s − 1.16·13^-s + 1.26i·17^-s − 1.48i·19^-s + 1.61·23^-s − 0.200·25^-s − 0.407i·29^-s − 1.16i·31^-s + 0.214·35^-s + 0.328·37^-s − 1.28i·41^-s − 1.77i·43^-s − 0.369·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2160$ = $2^{4} \cdot 3^{3} \cdot 5$
Sign:	$i$
Analytic conductor:	$17.2476$
Root analytic conductor:	$4.15303$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2160} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2160,\ (\ :1/2),\ i)$

Particular Values

$L(1)$	$\approx$	$1.186757767$
$L(\frac12)$	$\approx$	$1.186757767$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
good	7	$ 1 - 1.26iT - 7T^{2} $
	11	$ 1 + 1.26T + 11T^{2} $
	13	$ 1 + 4.19T + 13T^{2} $
	17	$ 1 - 5.19iT - 17T^{2} $
	19	$ 1 + 6.46iT - 19T^{2} $
	23	$ 1 - 7.73T + 23T^{2} $
	29	$ 1 + 2.19iT - 29T^{2} $
	31	$ 1 + 6.46iT - 31T^{2} $
	37	$ 1 - 2T + 37T^{2} $
	41	$ 1 + 8.19iT - 41T^{2} $
	43	$ 1 + 11.6iT - 43T^{2} $
	47	$ 1 + 2.53T + 47T^{2} $
	53	$ 1 - 0.803iT - 53T^{2} $
	59	$ 1 + 11.6T + 59T^{2} $
	61	$ 1 + T + 61T^{2} $
	67	$ 1 + 10.3iT - 67T^{2} $
	71	$ 1 + 1.26T + 71T^{2} $
	73	$ 1 - 6.19T + 73T^{2} $
	79	$ 1 + 3.92iT - 79T^{2} $
	83	$ 1 - 5.19T + 83T^{2} $
	89	$ 1 - 3.80iT - 89T^{2} $
	97	$ 1 + 8T + 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

−9.080122044530265463173610861053, −8.119946811515155234991151682870, −7.37471491380295744333542600270, −6.58163375634899606061836509029, −5.53847089177282888754189162777, −4.97232893074457751734717892370, −4.07932918198831638818367437533, −2.83995562186468823654378980564, −2.03031818208954932468058188193, −0.43631070961811189632071913433, 1.23032000481983733540545183292, 2.67201338497142055084619081595, 3.28590817834723512813050154802, 4.58802398679959866710061836122, 5.13870464228729663608012801838, 6.21719957552703635874923542516, 7.13276125225537578475590837934, 7.51924885595354749275959020052, 8.413996668302737145806694325773, 9.478725381417883425772747196160

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.186757767\)
\(L(\frac12)\)	\(\approx\)	\(1.186757767\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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