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Properties

Label 1-2883-2883.674-r0-0-0
Degree $1$
Conductor $2883$
Sign $0.998 - 0.0591i$
Analytic cond. $13.3885$
Root an. cond. $13.3885$
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  + (−0.595 − 0.803i)2-s + (−0.289 + 0.957i)4-s + (0.440 − 0.897i)5-s + (0.843 − 0.537i)7-s + (0.941 − 0.337i)8-s + (−0.983 + 0.181i)10-s + (0.111 + 0.993i)11-s + (0.703 + 0.710i)13-s + (−0.934 − 0.356i)14-s + (−0.832 − 0.554i)16-s + (0.191 + 0.981i)17-s + (−0.926 + 0.375i)19-s + (0.731 + 0.681i)20-s + (0.731 − 0.681i)22-s + (−0.989 + 0.141i)23-s + ⋯
L(s)  = 1  + (−0.595 − 0.803i)2-s + (−0.289 + 0.957i)4-s + (0.440 − 0.897i)5-s + (0.843 − 0.537i)7-s + (0.941 − 0.337i)8-s + (−0.983 + 0.181i)10-s + (0.111 + 0.993i)11-s + (0.703 + 0.710i)13-s + (−0.934 − 0.356i)14-s + (−0.832 − 0.554i)16-s + (0.191 + 0.981i)17-s + (−0.926 + 0.375i)19-s + (0.731 + 0.681i)20-s + (0.731 − 0.681i)22-s + (−0.989 + 0.141i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $0.998 - 0.0591i$
Analytic conductor: \(13.3885\)
Root analytic conductor: \(13.3885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2883,\ (0:\ ),\ 0.998 - 0.0591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.304354282 - 0.03860089453i\)
\(L(\frac12)\) \(\approx\) \(1.304354282 - 0.03860089453i\)
\(L(1)\) \(\approx\) \(0.8789824301 - 0.2640206593i\)
\(L(1)\) \(\approx\) \(0.8789824301 - 0.2640206593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.595 - 0.803i)T \)
5 \( 1 + (0.440 - 0.897i)T \)
7 \( 1 + (0.843 - 0.537i)T \)
11 \( 1 + (0.111 + 0.993i)T \)
13 \( 1 + (0.703 + 0.710i)T \)
17 \( 1 + (0.191 + 0.981i)T \)
19 \( 1 + (-0.926 + 0.375i)T \)
23 \( 1 + (-0.989 + 0.141i)T \)
29 \( 1 + (-0.171 + 0.985i)T \)
37 \( 1 + (-0.151 - 0.988i)T \)
41 \( 1 + (-0.0303 - 0.999i)T \)
43 \( 1 + (0.131 + 0.991i)T \)
47 \( 1 + (-0.884 - 0.467i)T \)
53 \( 1 + (0.385 + 0.922i)T \)
59 \( 1 + (0.703 + 0.710i)T \)
61 \( 1 + (0.994 + 0.101i)T \)
67 \( 1 + (-0.874 + 0.485i)T \)
71 \( 1 + (-0.970 + 0.240i)T \)
73 \( 1 + (0.965 - 0.260i)T \)
79 \( 1 + (0.965 + 0.260i)T \)
83 \( 1 + (0.595 + 0.803i)T \)
89 \( 1 + (-0.998 + 0.0607i)T \)
97 \( 1 + (0.796 + 0.604i)T \)
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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

−18.86534229072416937709320497650, −18.32815691998177948001243430266, −17.8374449383318223275032021171, −17.21148171671621486540701110635, −16.32777782617036752525414417644, −15.61207900357530627410027214834, −15.01157736755129545534853380196, −14.37172452648869462235972149928, −13.74245345971407719532291566955, −13.16150780410114814752004383425, −11.661009020647076219281668135, −11.25228358061869342964434446722, −10.472252818050353388423210740377, −9.83682809232014940103471953436, −8.92518769220621385355621877011, −8.239088325136899327656886868501, −7.75947931204234674655026403589, −6.678116687200966887790793782296, −6.1193749010676710895390661361, −5.523267730903776960678474290485, −4.68877953812910299801060002770, −3.52456007790004300840595502912, −2.50418987253890598802780490109, −1.71954981585837915666670045679, −0.543225405813763799128426391, 1.04349528768539090234470992209, 1.794800932577224208978837548085, 2.11292294558520803878071633791, 3.80988320571048263496409768124, 4.137619983610646605377143640157, 4.92388950974957229874575838864, 5.9650016505415139730506655831, 7.00728213234060838820933728650, 7.835514578658554292737248451583, 8.5389489083982195583228528440, 9.01342359147266995237818913107, 9.977951795260795182356081886550, 10.46761999312030567400942114419, 11.24752992871733340241514358030, 12.09141470575174654524954137845, 12.61562199437225872205658406489, 13.29410381082657221332642331132, 14.08436844757566788228678141661, 14.75876800492767249194569116116, 15.935430809229081497743322989028, 16.67766908390566026178503071931, 17.09296999566687748201911971243, 17.93670668717268248033132104579, 18.1436844386432310883391417267, 19.442238272820972206418907320907

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