Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{2} $
Sign $0.920 - 0.389i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.987 + 0.156i)3-s + 7-s + (0.951 + 0.309i)9-s + (0.453 − 0.891i)11-s + (0.891 − 0.453i)13-s + (0.587 − 0.809i)17-s + (0.987 − 0.156i)19-s + (0.987 + 0.156i)21-s + (0.309 + 0.951i)23-s + (0.891 + 0.453i)27-s + (−0.156 + 0.987i)29-s + (−0.809 − 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.453 − 0.891i)37-s + (0.951 − 0.309i)39-s + ⋯
L(s,χ)  = 1  + (0.987 + 0.156i)3-s + 7-s + (0.951 + 0.309i)9-s + (0.453 − 0.891i)11-s + (0.891 − 0.453i)13-s + (0.587 − 0.809i)17-s + (0.987 − 0.156i)19-s + (0.987 + 0.156i)21-s + (0.309 + 0.951i)23-s + (0.891 + 0.453i)27-s + (−0.156 + 0.987i)29-s + (−0.809 − 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.453 − 0.891i)37-s + (0.951 − 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.920 - 0.389i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.920 - 0.389i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(800\)    =    \(2^{5} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.920 - 0.389i$
motivic weight  =  \(0\)
character  :  $\chi_{800} (613, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 800,\ (1:\ ),\ 0.920 - 0.389i)$
$L(\chi,\frac{1}{2})$  $\approx$  $4.296679336 - 0.8722170815i$
$L(\frac12,\chi)$  $\approx$  $4.296679336 - 0.8722170815i$
$L(\chi,1)$  $\approx$  1.948809134 - 0.1225216154i
$L(1,\chi)$  $\approx$  1.948809134 - 0.1225216154i

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

−21.97258633888924086619177883754, −20.94711842568223977530224297169, −20.68621520616745063350602954357, −19.82075592294521083964189986091, −18.88361574952947650392578858560, −18.24112716936754389943712899608, −17.44053574602327860989354906594, −16.41354163073049728003606971964, −15.41680529233527012174349398096, −14.65452383665393937027305114135, −14.185466808476466699075024165623, −13.25757571667406909605975431029, −12.347540209461669351103134886663, −11.52346746307879653605633508545, −10.44360472494241799460012397960, −9.57135037730892327576165781535, −8.67577516803829836753192201115, −7.996705076736010069739868961603, −7.18444992301608740941056789953, −6.18683915420259189645948494560, −4.83633423018180076609570629803, −4.064179414023932989259315038389, −3.089615988838035006793777874244, −1.78266296799620717085022514457, −1.30132965944522993649462326907, 0.95096008521923028763303527404, 1.78584494941811773232759597137, 3.1726432610054793021511021154, 3.668496824482298235044260823978, 4.96500791327470258879205599211, 5.7406497601671849679211413518, 7.242655928745491936670253814531, 7.78175319357557877079419389326, 8.80907532042110661895007509334, 9.24971743716175194711014778141, 10.50891437819553019302917943357, 11.23122927735085023588072954865, 12.14027694721627680983196525564, 13.40662983407067579553550956116, 13.9007120885110670697170923666, 14.57521352938803842140322631184, 15.51199151338659835762859654069, 16.17269157149496334827880029405, 17.17670182379459821620519233526, 18.38296985069129921706772833559, 18.590242497590760951714275785524, 19.882352979588378957515591853710, 20.33907927000365711008006227259, 21.19846698227550346204276428048, 21.718019250072985127400202207828

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