LMFDB - L-function 2-2160-12.11-c1-0-22

Dirichlet series

L(s) = 1

− i·5^-s − 4.43i·7^-s + 6.16·11^-s + 13^-s + 4.68i·17^-s + 4.43i·19^-s + 6.16·23^-s − 25^-s − 10.6i·29^-s + 6.16i·31^-s − 4.43·35^-s + 9.68·37^-s − 6i·41^-s + 6.16i·43^-s + 4.22·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 1.67i·7^-s + 1.85·11^-s + 0.277·13^-s + 1.13i·17^-s + 1.01i·19^-s + 1.28·23^-s − 0.200·25^-s − 1.98i·29^-s + 1.10i·31^-s − 0.749·35^-s + 1.59·37^-s − 0.937i·41^-s + 0.940i·43^-s + 0.616·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2160$ = $2^{4} \cdot 3^{3} \cdot 5$
Sign:	$0.5 + 0.866i$
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:	$(2,\ 2160,\ (\ :1/2),\ 0.5 + 0.866i)$

Particular Values

$L(1)$	$\approx$	$2.081349039$
$L(\frac12)$	$\approx$	$2.081349039$
$L(\frac{3}{2})$		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 + 4.43iT - 7T^{2} $
	11	$ 1 - 6.16T + 11T^{2} $
	13	$ 1 - T + 13T^{2} $
	17	$ 1 - 4.68iT - 17T^{2} $
	19	$ 1 - 4.43iT - 19T^{2} $
	23	$ 1 - 6.16T + 23T^{2} $
	29	$ 1 + 10.6iT - 29T^{2} $
	31	$ 1 - 6.16iT - 31T^{2} $
	37	$ 1 - 9.68T + 37T^{2} $
	41	$ 1 + 6iT - 41T^{2} $
	43	$ 1 - 6.16iT - 43T^{2} $
	47	$ 1 - 4.22T + 47T^{2} $
	53	$ 1 + 6iT - 53T^{2} $
	59	$ 1 + 12.3T + 59T^{2} $
	61	$ 1 + 5.68T + 61T^{2} $
	67	$ 1 + 9.84iT - 67T^{2} $
	71	$ 1 + 1.94T + 71T^{2} $
	73	$ 1 + 3.68T + 73T^{2} $
	79	$ 1 - 0.213iT - 79T^{2} $
	83	$ 1 - 10.3T + 83T^{2} $
	89	$ 1 + 12iT - 89T^{2} $
	97	$ 1 - 5.68T + 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

−9.022849998100864786989041481330, −8.088466811598151972996359541252, −7.46551516219213884263274373409, −6.48928963243725179878264570806, −6.07337364659142444285787942387, −4.60895332732320357742003346088, −4.06186467172519733519678753701, −3.44276207779699811700313463128, −1.60896561322612134583112657139, −0.907049188436772176755654176204, 1.23176777202417811236291132437, 2.54953577563469058137524074230, 3.17539595545367624320711572808, 4.41360023799890722712487776359, 5.27802781759528911179920649060, 6.16163707479601462952481004512, 6.75300806434285253181871862109, 7.54505530639587003683804490674, 8.910002249795128943065250495601, 9.009385160514187914045177493477

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

\(L(1)\)	\(\approx\)	\(2.081349039\)
\(L(\frac12)\)	\(\approx\)	\(2.081349039\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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