Properties

Label 1-571-571.164-r1-0-0
Degree $1$
Conductor $571$
Sign $0.928 + 0.372i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (0.401 + 0.915i)2-s + (−0.945 − 0.324i)3-s + (−0.677 + 0.735i)4-s + (0.245 + 0.969i)5-s + (−0.0825 − 0.996i)6-s + (−0.546 − 0.837i)7-s + (−0.945 − 0.324i)8-s + (0.789 + 0.614i)9-s + (−0.789 + 0.614i)10-s + (0.789 + 0.614i)11-s + (0.879 − 0.475i)12-s + (−0.986 − 0.164i)13-s + (0.546 − 0.837i)14-s + (0.0825 − 0.996i)15-s + (−0.0825 − 0.996i)16-s + (0.0825 + 0.996i)17-s + ⋯
L(s)  = 1  + (0.401 + 0.915i)2-s + (−0.945 − 0.324i)3-s + (−0.677 + 0.735i)4-s + (0.245 + 0.969i)5-s + (−0.0825 − 0.996i)6-s + (−0.546 − 0.837i)7-s + (−0.945 − 0.324i)8-s + (0.789 + 0.614i)9-s + (−0.789 + 0.614i)10-s + (0.789 + 0.614i)11-s + (0.879 − 0.475i)12-s + (−0.986 − 0.164i)13-s + (0.546 − 0.837i)14-s + (0.0825 − 0.996i)15-s + (−0.0825 − 0.996i)16-s + (0.0825 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9048033929 + 0.1747978345i\)
\(L(\frac12)\) \(\approx\) \(0.9048033929 + 0.1747978345i\)
\(L(1)\) \(\approx\) \(0.6941746527 + 0.3738674527i\)
\(L(1)\) \(\approx\) \(0.6941746527 + 0.3738674527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.401 + 0.915i)T \)
3 \( 1 + (-0.945 - 0.324i)T \)
5 \( 1 + (0.245 + 0.969i)T \)
7 \( 1 + (-0.546 - 0.837i)T \)
11 \( 1 + (0.789 + 0.614i)T \)
13 \( 1 + (-0.986 - 0.164i)T \)
17 \( 1 + (0.0825 + 0.996i)T \)
19 \( 1 + (-0.945 - 0.324i)T \)
23 \( 1 + (0.945 - 0.324i)T \)
29 \( 1 + (-0.986 - 0.164i)T \)
31 \( 1 + (-0.0825 - 0.996i)T \)
37 \( 1 + (-0.986 + 0.164i)T \)
41 \( 1 + (-0.245 - 0.969i)T \)
43 \( 1 + (0.789 - 0.614i)T \)
47 \( 1 + (0.0825 + 0.996i)T \)
53 \( 1 + (0.401 - 0.915i)T \)
59 \( 1 + (0.546 + 0.837i)T \)
61 \( 1 + (-0.401 - 0.915i)T \)
67 \( 1 + (0.401 - 0.915i)T \)
71 \( 1 - T \)
73 \( 1 + (0.986 - 0.164i)T \)
79 \( 1 + (0.677 + 0.735i)T \)
83 \( 1 + (-0.0825 - 0.996i)T \)
89 \( 1 + (0.0825 + 0.996i)T \)
97 \( 1 + (-0.677 + 0.735i)T \)
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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

−22.76769729032869277705754610298, −22.00565454161947226646082395648, −21.4684545039721508953745409746, −20.7792416864055425905493389954, −19.64653868417939809507730244649, −19.00654317713006313693998027771, −18.033173039225601917956332660750, −17.06939082957542138389708350932, −16.43513483424118234726849737370, −15.39739443423788761826190103615, −14.463377727694712790276005217908, −13.26638684385571294296355018338, −12.503455799861786962185865386577, −11.97364169989298591787822143423, −11.22486725022692162790359342906, −10.06428567130030208058253624770, −9.3059623988995226517992063293, −8.78531746033848764101727953457, −6.84192924128898999176324794301, −5.774942698021046338806035012640, −5.19393780636775144550728059960, −4.33779237926389408155390364577, −3.20079631188499182394393255805, −1.85271660190015322473202376517, −0.72104256272716775624093494337, 0.33306040033122216336169433005, 2.12666386888105460684397816632, 3.65075743812036609955576467666, 4.45431298002657762129231180351, 5.63330430779840894379297071923, 6.55850845582913937525953931405, 6.98991488403832882602043416977, 7.71923484819247610797183385001, 9.31549491250636171657132073648, 10.22873322201539405824539190304, 11.05428128695050870420306964099, 12.285453765706476446079609267623, 12.9036311406733092164899108316, 13.79260026708094703792865263236, 14.80413997126933048438799771262, 15.32836301791460327542605439936, 16.676366333936043732651690713834, 17.24673552839687900506628847345, 17.53538565600528611847657153839, 18.87858377857087913649272792716, 19.35796259963471396144370928478, 20.917904750654542238367997799366, 22.01222815095062793497960846864, 22.48301798042146860900931615882, 22.950046875469658993445606263288

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