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Properties

Label 1-200-200.117-r1-0-0
Degree $1$
Conductor $200$
Sign $0.535 + 0.844i$
Analytic cond. $21.4929$
Root an. cond. $21.4929$
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  + (0.951 + 0.309i)3-s + i·7-s + (0.809 + 0.587i)9-s + (0.809 − 0.587i)11-s + (0.587 − 0.809i)13-s + (−0.951 + 0.309i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.587 + 0.809i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (0.309 + 0.951i)31-s + (0.951 − 0.309i)33-s + (−0.587 + 0.809i)37-s + (0.809 − 0.587i)39-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s + i·7-s + (0.809 + 0.587i)9-s + (0.809 − 0.587i)11-s + (0.587 − 0.809i)13-s + (−0.951 + 0.309i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.587 + 0.809i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (0.309 + 0.951i)31-s + (0.951 − 0.309i)33-s + (−0.587 + 0.809i)37-s + (0.809 − 0.587i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(21.4929\)
Root analytic conductor: \(21.4929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (1:\ ),\ 0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.463110196 + 1.354106289i\)
\(L(\frac12)\) \(\approx\) \(2.463110196 + 1.354106289i\)
\(L(1)\) \(\approx\) \(1.563299687 + 0.4059571238i\)
\(L(1)\) \(\approx\) \(1.563299687 + 0.4059571238i\)

Euler product

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

−26.418429381300804514711498472049, −25.6936058423339313392728981923, −24.6136638972101986981516788101, −23.87828873125293916611377851037, −22.83116559749745957926638024200, −21.66726868322195811061382161579, −20.44705845908870233224680652279, −20.034337716884524418737790653799, −19.022892897822038756016258583973, −17.959882710361031530300974666871, −16.93914130584798633179161319648, −15.759963619576055457769520891746, −14.68374452250836298077719737029, −13.80734656339698320057608914323, −13.12240207710949501301648791832, −11.80993098175239416258449948994, −10.602802266545454671990975651258, −9.34505264694143854227660049022, −8.63163650254542688860860180053, −7.11612914072150525427161479123, −6.747606308024740074423115826047, −4.57106888721334335398336912727, −3.73436456178155270251266321211, −2.260465844462986201750915411784, −0.959862229885284892341457941921, 1.50246371261601135219053491192, 2.883896140349653383895824231543, 3.84593012335570618154060264336, 5.32320751959546954083786991569, 6.5385072293250768067193691696, 8.10601262587672124911430831316, 8.745085332699347822191339803221, 9.72384655028192325696966101914, 10.95329723585305142101822989153, 12.16250099717785585119595612938, 13.31312197425891799390942257352, 14.19888573584250204923244113174, 15.32520539159188972185724739210, 15.7967841951915740175253496487, 17.20588687022093876986744324373, 18.44511690058578714121285180245, 19.23770401140920704002696974646, 20.12945482559508898191263016893, 21.1672344182091847602328278512, 21.89329294965575373431155297909, 22.86283235908218136145755993962, 24.40890331062476866970799036294, 24.99237717143557136041276960990, 25.70933712944488351756462916746, 26.932818131764090620950525681190

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