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Properties

Label 1-200-200.21-r0-0-0
Degree $1$
Conductor $200$
Sign $0.535 - 0.844i$
Analytic cond. $0.928796$
Root an. cond. $0.928796$
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  + (−0.309 − 0.951i)3-s + 7-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.309 − 0.951i)21-s + (−0.809 − 0.587i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)33-s + (0.809 − 0.587i)37-s + (−0.809 − 0.587i)39-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + 7-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.309 − 0.951i)21-s + (−0.809 − 0.587i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)33-s + (0.809 − 0.587i)37-s + (−0.809 − 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(0.928796\)
Root analytic conductor: \(0.928796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (0:\ ),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.040693357 - 0.5721260147i\)
\(L(\frac12)\) \(\approx\) \(1.040693357 - 0.5721260147i\)
\(L(1)\) \(\approx\) \(1.032624294 - 0.3319573545i\)
\(L(1)\) \(\approx\) \(1.032624294 - 0.3319573545i\)

Euler product

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

−27.16648536375092136918154233581, −26.26319024656322280711311972661, −25.327939929239203759111454250542, −23.92438971402613971126642870101, −23.46240934071449897012221980182, −21.80057543319904379808371179324, −21.72745084588767567378441814007, −20.54416147113324436426753536110, −19.608846609683568961856634849881, −18.26935404298679345531803455744, −17.273854273843435924379402119545, −16.53129423109257270860148445048, −15.4267173826685565637825413485, −14.5514829134894910930326577621, −13.65940039807263016811371895021, −11.95857912684032758683550202463, −11.243920251523204464556676901985, −10.390064341480197906218815357206, −9.017827924832673812419031925172, −8.35762918882993024295670316128, −6.62409005725283799438506788200, −5.53166484364844090124808636854, −4.38736573001378531112502530712, −3.4508369809728700803769304941, −1.51913789710526858107221771566, 1.16678896088193981694731306773, 2.31006237250729210363208505240, 4.10181513124177438500224428307, 5.46778875770818886817227743445, 6.47266908409354110707482063383, 7.71404037427088751884778188494, 8.396610716358763270866166752734, 9.93853993881460957367553307906, 11.30802136675975427035154910783, 11.90187025148506168750763510395, 13.022889194788427487500870169461, 14.08060849347568135755753762639, 14.864947711313681533897428250826, 16.3599073387843128993688030911, 17.352760247028595610641327830618, 18.12694528527520289282473158893, 18.86802489067200219529053873949, 20.18261053979356186310876355564, 20.81074083548657058008753405540, 22.32162199326546611890079520802, 23.02904895817732906152786642958, 23.94730394736278481492288779724, 24.94998306851510643653632825282, 25.37009877295466429648652677538, 26.88877084334063813598467113632

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