LMFDB - L-function 2-2160-12.11-c3-0-54

Dirichlet series

L(s) = 1

− 5i·5^-s − 15.0i·7^-s + 42.5·11^-s − 43.6·13^-s − 39.0i·17^-s + 43.1i·19^-s + 151.·23^-s − 25·25^-s + 118. i·29^-s + 270. i·31^-s − 75.1·35^-s + 202.·37^-s − 95.4i·41^-s + 47.6i·43^-s + 236.·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 0.811i·7^-s + 1.16·11^-s − 0.931·13^-s − 0.557i·17^-s + 0.520i·19^-s + 1.37·23^-s − 0.200·25^-s + 0.760i·29^-s + 1.56i·31^-s − 0.362·35^-s + 0.901·37^-s − 0.363i·41^-s + 0.169i·43^-s + 0.735·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2160$ = $2^{4} \cdot 3^{3} \cdot 5$
Sign:	$0.866 + 0.5i$
Analytic conductor:	$127.444$
Root analytic conductor:	$11.2891$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2160} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2160,\ (\ :3/2),\ 0.866 + 0.5i)$

Particular Values

$L(2)$	$\approx$	$2.356037300$
$L(\frac12)$	$\approx$	$2.356037300$
$L(\frac{5}{2})$		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 + 5iT $
good	7	$ 1 + 15.0iT - 343T^{2} $
	11	$ 1 - 42.5T + 1.33e3T^{2} $
	13	$ 1 + 43.6T + 2.19e3T^{2} $
	17	$ 1 + 39.0iT - 4.91e3T^{2} $
	19	$ 1 - 43.1iT - 6.85e3T^{2} $
	23	$ 1 - 151.T + 1.21e4T^{2} $
	29	$ 1 - 118. iT - 2.43e4T^{2} $
	31	$ 1 - 270. iT - 2.97e4T^{2} $
	37	$ 1 - 202.T + 5.06e4T^{2} $
	41	$ 1 + 95.4iT - 6.89e4T^{2} $
	43	$ 1 - 47.6iT - 7.95e4T^{2} $
	47	$ 1 - 236.T + 1.03e5T^{2} $
	53	$ 1 - 227. iT - 1.48e5T^{2} $
	59	$ 1 + 335.T + 2.05e5T^{2} $
	61	$ 1 - 324.T + 2.26e5T^{2} $
	67	$ 1 - 161. iT - 3.00e5T^{2} $
	71	$ 1 + 451.T + 3.57e5T^{2} $
	73	$ 1 - 315.T + 3.89e5T^{2} $
	79	$ 1 + 753. iT - 4.93e5T^{2} $
	83	$ 1 - 554.T + 5.71e5T^{2} $
	89	$ 1 - 739. iT - 7.04e5T^{2} $
	97	$ 1 - 1.71e3T + 9.12e5T^{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

−8.929478006699708268516990194021, −7.73470738010756740351692475418, −7.12823003612717933154439537181, −6.48255170082578288064100118945, −5.31597024285308790883776875534, −4.64376718692318303721968502244, −3.81123644768563459403677756278, −2.85126959339786004345399664929, −1.48479360903128857355474273489, −0.72139503586435457736128591431, 0.72692362387743227038536098835, 2.07820439060660618932795897906, 2.80324344900558349119279936374, 3.89813246604426083697872362960, 4.74515171740814341058404574765, 5.74850268701289563157359131084, 6.42021113806954393541067829363, 7.19188069824985285194892143977, 7.978388519699883442649797033112, 8.976169519210912406646627145765

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

\(L(2)\)	\(\approx\)	\(2.356037300\)
\(L(\frac12)\)	\(\approx\)	\(2.356037300\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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