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
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
$p$ | $F_p(T)$ | |
---|---|---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (0.401 + 0.915i)T \) |
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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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