LMFDB - L-function 2-2160-12.11-c1-0-9

Dirichlet series

L(s) = 1

− i·5^-s + 4.73i·7^-s − 4.73·11^-s + 6.19·13^-s − 5.19i·17^-s + 0.464i·19^-s + 4.26·23^-s − 25^-s + 8.19i·29^-s + 0.464i·31^-s + 4.73·35^-s + 2·37^-s + 2.19i·41^-s + 5.66i·43^-s − 9.46·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s + 1.78i·7^-s − 1.42·11^-s + 1.71·13^-s − 1.26i·17^-s + 0.106i·19^-s + 0.889·23^-s − 0.200·25^-s + 1.52i·29^-s + 0.0833i·31^-s + 0.799·35^-s + 0.328·37^-s + 0.342i·41^-s + 0.863i·43^-s − 1.38·47^-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

$L(1)$	$\approx$	$1.420906987$
$L(\frac12)$	$\approx$	$1.420906987$
$L(\frac{3}{2})$		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 - 4.73iT - 7T^{2} $
	11	$ 1 + 4.73T + 11T^{2} $
	13	$ 1 - 6.19T + 13T^{2} $
	17	$ 1 + 5.19iT - 17T^{2} $
	19	$ 1 - 0.464iT - 19T^{2} $
	23	$ 1 - 4.26T + 23T^{2} $
	29	$ 1 - 8.19iT - 29T^{2} $
	31	$ 1 - 0.464iT - 31T^{2} $
	37	$ 1 - 2T + 37T^{2} $
	41	$ 1 - 2.19iT - 41T^{2} $
	43	$ 1 - 5.66iT - 43T^{2} $
	47	$ 1 + 9.46T + 47T^{2} $
	53	$ 1 - 11.1iT - 53T^{2} $
	59	$ 1 - 5.66T + 59T^{2} $
	61	$ 1 + T + 61T^{2} $
	67	$ 1 - 10.3iT - 67T^{2} $
	71	$ 1 + 4.73T + 71T^{2} $
	73	$ 1 + 4.19T + 73T^{2} $
	79	$ 1 - 9.92iT - 79T^{2} $
	83	$ 1 + 5.19T + 83T^{2} $
	89	$ 1 - 14.1iT - 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

−9.052813649819912957001489825625, −8.605120203779839910341464199134, −7.968977223843871554604815401697, −6.88751746539785423955840514277, −5.87922755984119348496344402269, −5.37626723346144320415770465442, −4.69909309616188821483787809321, −3.18986345766012403224163924120, −2.62688782480522439807356017849, −1.31398808466998804631050492899, 0.52576368104977398623015708558, 1.80484765349897089130126594125, 3.24554581397598036876368584785, 3.84678764616181897334961248691, 4.72271820736228343799453897253, 5.88089786954278546467640145991, 6.54801771529231945686448002656, 7.41428225271384938875282564884, 8.019484424208002091569181280772, 8.672676622884276120883968149444

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.420906987\)
\(L(\frac12)\)	\(\approx\)	\(1.420906987\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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