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Properties

Label 1-800-800.91-r1-0-0
Degree $1$
Conductor $800$
Sign $-0.920 + 0.389i$
Analytic cond. $85.9719$
Root an. cond. $85.9719$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.156 + 0.987i)3-s − i·7-s + (−0.951 − 0.309i)9-s + (−0.453 + 0.891i)11-s + (−0.453 − 0.891i)13-s + (0.809 + 0.587i)17-s + (0.987 − 0.156i)19-s + (0.987 + 0.156i)21-s + (−0.951 + 0.309i)23-s + (0.453 − 0.891i)27-s + (0.156 − 0.987i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.891 − 0.453i)37-s + (0.951 − 0.309i)39-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)3-s − i·7-s + (−0.951 − 0.309i)9-s + (−0.453 + 0.891i)11-s + (−0.453 − 0.891i)13-s + (0.809 + 0.587i)17-s + (0.987 − 0.156i)19-s + (0.987 + 0.156i)21-s + (−0.951 + 0.309i)23-s + (0.453 − 0.891i)27-s + (0.156 − 0.987i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.891 − 0.453i)37-s + (0.951 − 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.920 + 0.389i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.920 + 0.389i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1530152474 + 0.7537773171i\)
\(L(\frac12)\) \(\approx\) \(0.1530152474 + 0.7537773171i\)
\(L(1)\) \(\approx\) \(0.8305561665 + 0.2405364691i\)
\(L(1)\) \(\approx\) \(0.8305561665 + 0.2405364691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.156 + 0.987i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.453 + 0.891i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.891 - 0.453i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (-0.891 + 0.453i)T \)
61 \( 1 + (-0.891 - 0.453i)T \)
67 \( 1 + (-0.987 + 0.156i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.987 - 0.156i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−21.84359267382675652389368402628, −20.96645430702970732716710389889, −19.94460364323675132567479873190, −19.11322249421134696900554622271, −18.389083866932501034696260449867, −18.1449822572075318070646670017, −16.73812518432406956985563780421, −16.31649177744366146111850403037, −15.17944432729089607974491108561, −14.1140049416953606799633490248, −13.70198650195721288196071302740, −12.55052332278952566652265923930, −11.87785652851382308334130801799, −11.409360443953170906979152612479, −10.08399100362960966683910965653, −9.08437894796697463910999101090, −8.240372786649462875223202530618, −7.498460844575584538016214933846, −6.43695012284555258285519877765, −5.6993147541606020496848786293, −4.90993295715665057564347362719, −3.24910919285742803174297019877, −2.490764643792277666735048480253, −1.429688121632423409363606479134, −0.195670706938623286684802013524, 1.04719906063432535757638776284, 2.64416286205913534587195811605, 3.60491031053946701856417389086, 4.476884565215132334035364026899, 5.26846459761376794415084547402, 6.23102733062675978777908058655, 7.567937649457518766897093018394, 8.04379564641205903284418055026, 9.524442674644883716515245373918, 10.066320028260553023775766935105, 10.59084623206306621733591687234, 11.68929129351624032694529345860, 12.511639894760973538518585654717, 13.60302691662756883634308440385, 14.39493455724148220947373076256, 15.23908254491941599901182985630, 15.90045170990458170662913016220, 16.79362184635932934232360278910, 17.4895217884085487364708926668, 18.10836550623824429341425616518, 19.63002569875057541228681583155, 20.04759488938657420402135537267, 20.840488022225735949792606841071, 21.50678301564948718508408068516, 22.61071611301282125806613741956

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