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Properties

Label 1-569-569.169-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.146 + 0.989i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (0.0663 + 0.997i)2-s + (0.666 + 0.745i)3-s + (−0.991 + 0.132i)4-s + (−0.787 − 0.616i)5-s + (−0.699 + 0.714i)6-s + (−0.283 − 0.958i)7-s + (−0.197 − 0.980i)8-s + (−0.110 + 0.993i)9-s + (0.562 − 0.826i)10-s + (−0.903 − 0.428i)11-s + (−0.759 − 0.650i)12-s + (0.862 + 0.506i)13-s + (0.937 − 0.346i)14-s + (−0.0663 − 0.997i)15-s + (0.964 − 0.262i)16-s + (0.996 + 0.0883i)17-s + ⋯
L(s)  = 1  + (0.0663 + 0.997i)2-s + (0.666 + 0.745i)3-s + (−0.991 + 0.132i)4-s + (−0.787 − 0.616i)5-s + (−0.699 + 0.714i)6-s + (−0.283 − 0.958i)7-s + (−0.197 − 0.980i)8-s + (−0.110 + 0.993i)9-s + (0.562 − 0.826i)10-s + (−0.903 − 0.428i)11-s + (−0.759 − 0.650i)12-s + (0.862 + 0.506i)13-s + (0.937 − 0.346i)14-s + (−0.0663 − 0.997i)15-s + (0.964 − 0.262i)16-s + (0.996 + 0.0883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.146 + 0.989i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.146 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8597387878 + 0.9961181075i\)
\(L(\frac12)\) \(\approx\) \(0.8597387878 + 0.9961181075i\)
\(L(1)\) \(\approx\) \(0.8932999060 + 0.6052923010i\)
\(L(1)\) \(\approx\) \(0.8932999060 + 0.6052923010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.0663 + 0.997i)T \)
3 \( 1 + (0.666 + 0.745i)T \)
5 \( 1 + (-0.787 - 0.616i)T \)
7 \( 1 + (-0.283 - 0.958i)T \)
11 \( 1 + (-0.903 - 0.428i)T \)
13 \( 1 + (0.862 + 0.506i)T \)
17 \( 1 + (0.996 + 0.0883i)T \)
19 \( 1 + (0.991 + 0.132i)T \)
23 \( 1 + (0.525 + 0.850i)T \)
29 \( 1 + (-0.996 + 0.0883i)T \)
31 \( 1 + (0.921 + 0.387i)T \)
37 \( 1 + (0.283 + 0.958i)T \)
41 \( 1 + (-0.525 - 0.850i)T \)
43 \( 1 + (0.0663 - 0.997i)T \)
47 \( 1 + (0.999 - 0.0442i)T \)
53 \( 1 + (-0.699 + 0.714i)T \)
59 \( 1 + (0.999 - 0.0442i)T \)
61 \( 1 + (0.408 - 0.912i)T \)
67 \( 1 + (0.996 - 0.0883i)T \)
71 \( 1 + (-0.197 + 0.980i)T \)
73 \( 1 + (0.991 + 0.132i)T \)
79 \( 1 + (-0.0221 + 0.999i)T \)
83 \( 1 + (0.839 - 0.544i)T \)
89 \( 1 + (0.921 - 0.387i)T \)
97 \( 1 + (-0.240 - 0.970i)T \)
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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

−22.96764516477314527214879272694, −22.31987787561014968130800803247, −21.0174782355045414879924255582, −20.568893192355794328248430333795, −19.60696244741609147548803327183, −18.79984994660558022023742060999, −18.450378262124557143286857514176, −17.818889384378973341184475480565, −16.09920122705660587425686935977, −15.1131671380563668519039716028, −14.54927081438574048079718550424, −13.4093014965157599250601258307, −12.72770444462881507818183933623, −11.998128593407437119714985172175, −11.220101415351286495027810546705, −10.122314556980298933710644315506, −9.20670296314370602023787859668, −8.17984628700908974907346338286, −7.65674819646280543667563660494, −6.25470322183566641612520108686, −5.17879604835140937197490965376, −3.69862512276637601182265982694, −2.97666188983846756884767385953, −2.32331974176722982454050088222, −0.83947579636469679101348610672, 1.02224028203918055343725649690, 3.39542956781987600483047187207, 3.72990697993975146992421915686, 4.86348308495929095607477233088, 5.59475544044302135819325078595, 7.15982163404647518434942389518, 7.82928828001137942462787552035, 8.53157459832767904830595653832, 9.45910842364664874278087341942, 10.30808160090386328391806913257, 11.41284913110870477677870148442, 12.80488284884319104282835938985, 13.65013394795869952754891904076, 14.105954306941257130114971544163, 15.44866057876384239665179879588, 15.77565393217600023744927409870, 16.57845777712076715712640864653, 17.11412551487409397137066880044, 18.7051688102706981673550912883, 19.1338150603768405735473038284, 20.40856489542142241448640461293, 20.83672485690316773459620357224, 21.89291314596622123299081595270, 22.9651951708062615231476734288, 23.54367632367001664541390473120

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