LMFDB - L-function 1-2652-2652.395-r1-0-0

Dirichlet series

L(s) = 1

+ 5^-s + 7^-s − 11^-s − i·19^-s − i·23^-s + 25^-s + i·29^-s + 31^-s + 35^-s + 37^-s − 41^-s − 43^-s + i·47^-s + 49^-s + 53^-s + ⋯

L(s) = 1

+ 5^-s + 7^-s − 11^-s − i·19^-s − i·23^-s + 25^-s + i·29^-s + 31^-s + 35^-s + 37^-s − 41^-s − 43^-s + i·47^-s + 49^-s + 53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$2652$ = $2^{2} \cdot 3 \cdot 13 \cdot 17$
Sign:	$0.817 - 0.576i$
Analytic conductor:	$284.996$
Root analytic conductor:	$284.996$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2652} (395, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2652,\ (1:\ ),\ 0.817 - 0.576i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$3.077325434 - 0.9753897482i$
$L(\frac12)$	$\approx$	$3.077325434 - 0.9753897482i$
$L(1)$	$\approx$	$1.419023167 - 0.1044389049i$
$L(1)$	$\approx$	$1.419023167 - 0.1044389049i$

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	13	$ 1 $
	17	$ 1 $
good	5	$ 1 $
	7	$ 1 $
	11	$ 1 $
	19	$ 1 + T $
	23	$ 1 $
	29	$ 1 $
	31	$ 1 $
	37	$ 1 - T $
	41	$ 1 $
	43	$ 1 $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 $
	67	$ 1 $
	71	$ 1 - iT $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 - iT $
	97	$ 1 $
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$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−19.06628124778833041194069766910, −18.20100355742254896590927895820, −18.05159929479760485630625582068, −17.032124995857403102619128582015, −16.68924364869450604307074256716, −15.46970919461218609912385221975, −15.04593610460088530518310941344, −14.09042101633933530986709620524, −13.59690028320021256724456068464, −12.98422909567216806413507866090, −11.95385964878357504729290914003, −11.37875636869000991443021703775, −10.29974199217380175428004172490, −10.09724431551567492912727906742, −9.078612435732969820644683855265, −8.17533375555306330492883726612, −7.734479251923123947204970359893, −6.66617333141309780906204851451, −5.775834644468571528090022717061, −5.25452256837472487874686609911, −4.49678196410676019705481123618, −3.39201538091725265262349993431, −2.3607675132172796989856448361, −1.78625140591160202577216932942, −0.82975383726068013242808366206, 0.59604421040189945626982118941, 1.54173594423882186746546279586, 2.41708837262164797093758938372, 2.99738654077705976813935277490, 4.483520945731498206043359365095, 4.957468603207158159531910592367, 5.68027699916467590552363449423, 6.58402010919628939682821142314, 7.33914600997890928642900872942, 8.3269472695239181182872027387, 8.772617316735303994196935575551, 9.80177077593696269958540128631, 10.452148329940746585188898528312, 11.04104403041072449403684776798, 11.876622981609539590899196087952, 12.84863392721694465072679799273, 13.3717988401198282822783735328, 14.09470222571040427005868407060, 14.77238842847056209788693699857, 15.4310646951012477505647543336, 16.38190443444754104593581355994, 17.04026291405514442826242689063, 17.853328319119272544826586945853, 18.15681263052963023656045280019, 18.86969778321953187371429504729

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(3.077325434 - 0.9753897482i\)
\(L(\frac12)\)	\(\approx\)	\(3.077325434 - 0.9753897482i\)
\(L(1)\)	\(\approx\)	\(1.419023167 - 0.1044389049i\)
\(L(1)\)	\(\approx\)	\(1.419023167 - 0.1044389049i\)

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