Properties

Label 1-200-200.21-r0-0-0
Degree 11
Conductor 200200
Sign 0.5350.844i0.535 - 0.844i
Analytic cond. 0.9287960.928796
Root an. cond. 0.9287960.928796
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(200s/2ΓR(s)L(s)=((0.5350.844i)Λ(1s)\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}
Λ(s)=(200s/2ΓR(s)L(s)=((0.5350.844i)Λ(1s)\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: 11
Conductor: 200200    =    23522^{3} \cdot 5^{2}
Sign: 0.5350.844i0.535 - 0.844i
Analytic conductor: 0.9287960.928796
Root analytic conductor: 0.9287960.928796
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ200(21,)\chi_{200} (21, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 200, (0: ), 0.5350.844i)(1,\ 200,\ (0:\ ),\ 0.535 - 0.844i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.0406933570.5721260147i1.040693357 - 0.5721260147i
L(12)L(\frac12) \approx 1.0406933570.5721260147i1.040693357 - 0.5721260147i
L(1)L(1) \approx 1.0326242940.3319573545i1.032624294 - 0.3319573545i
L(1)L(1) \approx 1.0326242940.3319573545i1.032624294 - 0.3319573545i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
7 1+T 1 + T
11 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
13 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
17 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
19 1+(0.309+0.951i)T 1 + (-0.309 + 0.951i)T
23 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
29 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
31 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
37 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
41 1+(0.809+0.587i)T 1 + (-0.809 + 0.587i)T
43 1T 1 - T
47 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
53 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
59 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
61 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
67 1+(0.309+0.951i)T 1 + (-0.309 + 0.951i)T
71 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
73 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
79 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
83 1+(0.309+0.951i)T 1 + (-0.309 + 0.951i)T
89 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
97 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line