Properties

Degree 1
Conductor $ 2^{3} \cdot 5^{2} $
Sign $0.728 + 0.684i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(200\)    =    \(2^{3} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.728 + 0.684i$
motivic weight  =  \(0\)
character  :  $\chi_{200} (11, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 200,\ (1:\ ),\ 0.728 + 0.684i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6860868050 + 0.2716409826i$
$L(\frac12,\chi)$  $\approx$  $0.6860868050 + 0.2716409826i$
$L(\chi,1)$  $\approx$  0.6814058510 - 0.07659837243i
$L(1,\chi)$  $\approx$  0.6814058510 - 0.07659837243i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−26.53841919788634403076412866962, −25.846305122687593705659018652848, −24.59733734282868088013148455846, −23.48980959307803587926947436327, −22.65574759056682106239831203277, −22.04171482624800808507860329267, −20.98641315514544683332839454155, −19.91781802464345830480172394049, −18.902800412505651026853253307035, −17.67698930853866646790042660409, −16.92602978924085534978880507598, −15.93288503722467483487062645462, −15.222632334536307364482586924173, −13.88763347462555129128394338606, −12.556384495775575992116485667075, −11.87880225448825717449256085510, −10.6233834898413395527458419020, −9.7308501465721051223107856890, −8.88964936592152505579734742480, −6.86898901385480687064989915748, −6.44640655356585506865450204559, −4.82951134926387359103690088829, −4.06664121275498940259478793478, −2.40344475569585094997539836301, −0.36662623518164172743536726122, 0.91674788837959236660351520464, 2.61959204790307374855118323208, 4.081049392230957735098674625951, 5.720839284720482537365765982683, 6.286528051360041551960199058615, 7.50665550608588924350592105713, 8.706765877371283957971846331272, 10.15285252246408814557110688527, 10.97624159066657532212541060328, 12.19538113293942731914972964528, 12.95718510657323198512186802891, 13.83154359550680770659144476708, 15.37381823254475236346257861900, 16.31458501508164167395627697048, 17.18532970241603721276233468171, 18.073796583941104309591082577058, 19.299338345887763312989432602177, 19.649091314200291735584226907164, 21.37435888424012671988829229976, 22.2345373860722292347318253919, 22.94674680834010112566954539873, 23.916592605829102554102459627271, 24.81651437625586583531248089583, 25.66682772646407448652951735472, 26.907098482824209371178760932184

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