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

• Label: 1-2652-2652.395-r1-0-0
• Degree: $1$
• Conductor: $2652$
• Sign: $0.817 - 0.576i$
• Analytic cond.: $284.996$
• Root an. cond.: $284.996$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

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 + ⋯

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

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

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	$1$

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

Graph of the $Z$-function along the critical line