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Properties

Label 1-200-200.189-r0-0-0
Degree $1$
Conductor $200$
Sign $0.728 + 0.684i$
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  + (−0.809 − 0.587i)3-s − 7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (0.809 + 0.587i)21-s + (−0.309 + 0.951i)23-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.809 + 0.587i)31-s + (0.809 − 0.587i)33-s + (0.309 + 0.951i)37-s + (0.309 − 0.951i)39-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s − 7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (0.809 + 0.587i)21-s + (−0.309 + 0.951i)23-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.809 + 0.587i)31-s + (0.809 − 0.587i)33-s + (0.309 + 0.951i)37-s + (0.309 − 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(0.928796\)
Root analytic conductor: \(0.928796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (0:\ ),\ 0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6584581100 + 0.2607020083i\)
\(L(\frac12)\) \(\approx\) \(0.6584581100 + 0.2607020083i\)
\(L(1)\) \(\approx\) \(0.7449014455 + 0.03663100543i\)
\(L(1)\) \(\approx\) \(0.7449014455 + 0.03663100543i\)

Euler product

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

−26.819568329008114967322619521156, −26.04692169460539189179596716900, −24.90356153620592605387161982459, −23.74517139200526056452814740814, −22.85151353513186660037285593691, −22.21497803347987698254416850154, −21.221437967370332388828054738074, −20.280940293876777121239029353, −19.008895619532592619248525693259, −18.162742152717915667640739487924, −16.94725633167743130823456461472, −16.19258959086644361818143893320, −15.512955731890227315372443188581, −14.177728249678675502660918665826, −12.8734437671155039288789822882, −12.10773892986158363146096015517, −10.75558308458806596191268034188, −10.16693755988193574719605558141, −9.004093982703361799425873546049, −7.62875573310348619629975840891, −6.04754974585284364812676138723, −5.6697558423319968298568260335, −3.99307584527972927002000336918, −3.04328299018363114386013302345, −0.668666800054030249355587573933, 1.37562245064530140316742206219, 2.9405365599738725500207808725, 4.5803563197466120454355416724, 5.73284235859334558369929505025, 6.84581643776585274304993053165, 7.54877429531894697963579163712, 9.278524311103784235748572461546, 10.16671378867593656571000815042, 11.46230024336651007471289236097, 12.28940954659417670253247095291, 13.1858863647229139131505076754, 14.16004583308691683113357116985, 15.80360335550142519434745632140, 16.33809395326307264930350937806, 17.51273932680450656812465034506, 18.36067852937815920880330014968, 19.233244658637582352757247917779, 20.20784066527287104956499163471, 21.57368733199810978424091458414, 22.43895689493626941070174971712, 23.31189683204142296372102896376, 23.90242930364684375756204169785, 25.2843897567042795439285625248, 25.79231359244312646289418707220, 27.13191548863017571497407140628

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