Properties

Degree 1
Conductor $ 2^{4} \cdot 5^{2} $
Sign $-0.780 + 0.625i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.587 + 0.809i)3-s + 7-s + (−0.309 + 0.951i)9-s + (−0.951 + 0.309i)11-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)17-s + (−0.587 + 0.809i)19-s + (0.587 + 0.809i)21-s + (0.309 + 0.951i)23-s + (−0.951 + 0.309i)27-s + (−0.587 − 0.809i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.951 − 0.309i)37-s + (0.309 + 0.951i)39-s + ⋯
L(s,χ)  = 1  + (0.587 + 0.809i)3-s + 7-s + (−0.309 + 0.951i)9-s + (−0.951 + 0.309i)11-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)17-s + (−0.587 + 0.809i)19-s + (0.587 + 0.809i)21-s + (0.309 + 0.951i)23-s + (−0.951 + 0.309i)27-s + (−0.587 − 0.809i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.951 − 0.309i)37-s + (0.309 + 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.780 + 0.625i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.780 + 0.625i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(400\)    =    \(2^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.780 + 0.625i$
motivic weight  =  \(0\)
character  :  $\chi_{400} (291, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 400,\ (1:\ ),\ -0.780 + 0.625i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6590829957 + 1.876793585i$
$L(\frac12,\chi)$  $\approx$  $0.6590829957 + 1.876793585i$
$L(\chi,1)$  $\approx$  1.136419788 + 0.5859703972i
$L(1,\chi)$  $\approx$  1.136419788 + 0.5859703972i

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

−24.06301611425796014191781319356, −23.26392173215542787805897361849, −22.050578044884281940722732027002, −20.78171613110023603152594400638, −20.63032358395884614951632225952, −19.35670247911301037440521195712, −18.552078498640568876917636986470, −17.88008380271264947417254072042, −17.07220317026594921400076358425, −15.58043632183176842518313492195, −15.00845733597468985644670760689, −13.87330480860667848459234040829, −13.26581264930422512348857683555, −12.383766973634520175628367909364, −11.17144791518611797900919523835, −10.52932213033332241252218872857, −8.72672184697644773723111564527, −8.50314122617911573673434539003, −7.40953569749568132739446186342, −6.409227539149994664005949834384, −5.27157572962696071551739346670, −4.00091992034167989135015441605, −2.699745684298368992351367806282, −1.77208120262028143445527073189, −0.47313283487265654198998969202, 1.61945205274180900719862339667, 2.69675209388013321753493058212, 3.98342958751416423362913822880, 4.79345533167458378057558050205, 5.78120136159124075008786495548, 7.36670830774525717763835799930, 8.25667772753924291922564168498, 8.97858138229152740541231142079, 10.1482438584970648275406350065, 10.91280384175141130117758359877, 11.74745310111894839759386961866, 13.28172132680677153127621445544, 13.86043404704239194080807260622, 14.96010173823795170733571240600, 15.5384866377551708491702997955, 16.42954789885982663103643103231, 17.53072027526684036031767324179, 18.38703197902867878072083925547, 19.371902400078122943627210413525, 20.499017049095678199293680160200, 20.97889459172607760972066040111, 21.56815351954425221041427843293, 22.8276088017833366061060426958, 23.53509797895565640357348560698, 24.68562956453871279182998983124

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