↑

Properties

Label 1-200-200.133-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

Related objects

Downloads

Learn more

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: $\chi_{200} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (1:\ ),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4776918431 - 0.8689182369i\)
\(L(\frac12)\) \(\approx\) \(0.4776918431 - 0.8689182369i\)
\(L(1)\) \(\approx\) \(0.7584335525 - 0.2598860495i\)
\(L(1)\) \(\approx\) \(0.7584335525 - 0.2598860495i\)

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 \( 1 + (-0.951 - 0.309i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.46690384845051527331569638615, −25.995845917961699760588392390053, −25.042133605612400798132031768573, −24.1151696621129203949281771774, −23.08225830833803131506176453605, −22.10050091501912601018430951624, −21.71836795285528554019163526837, −20.39609580354128825448723554730, −19.291608788840644891739598029984, −18.10557691938082452955600550464, −17.46866268874571500832220322949, −16.435299037225759939176773983509, −15.36484048945690899077817421814, −14.71237170387386884074860726420, −13.003207947031023668431834825780, −12.09098099431067326721275202352, −11.4474303141550583423167853450, −10.04366690033533825420484344823, −9.35709789078849322709644443151, −7.8296895316480554493642736831, −6.528914603168548698084257131650, −5.541197528474537817786453173896, −4.611555880922160316863940104208, −3.07583786074432147356480083639, −1.338028220718576145683369649322, 0.426093457476797117680379725, 1.652057503606055424573902285255, 3.68535977492149538408978600236, 4.77802909836980375218196670071, 6.09987119348361775280738012952, 6.97929743273277697011116441150, 8.02458863165358880300425082731, 9.68753734023114365893392746474, 10.539301361188693144633041343449, 11.690234363984286298181617425037, 12.34290167282399151262949310592, 13.75494928659919674270362166385, 14.39514725654457349786834032260, 16.2014745041408947070759143809, 16.688152221721629838961306188459, 17.51463163807554682886416657651, 18.71777245416432873309501606172, 19.46109001252396864979476990593, 20.728753320112477371336947898061, 21.770886040222458973750250704197, 22.69661597706292567996391904260, 23.46512974650549962503980122374, 24.32219757057815057709118351954, 25.17369967310163796465377574271, 26.7336573239338550256616322965

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