Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.455 + 0.890i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.562 − 0.827i)3-s + (−0.809 + 0.587i)7-s + (−0.368 + 0.929i)9-s + (−0.860 + 0.509i)11-s + (−0.917 + 0.397i)13-s + (−0.481 − 0.876i)17-s + (−0.562 + 0.827i)19-s + (0.940 + 0.338i)21-s + (0.0627 + 0.998i)23-s + (0.975 − 0.218i)27-s + (−0.790 − 0.612i)29-s + (0.876 − 0.481i)31-s + (0.904 + 0.425i)33-s + (−0.218 + 0.975i)37-s + (0.844 + 0.535i)39-s + ⋯
L(s,χ)  = 1  + (−0.562 − 0.827i)3-s + (−0.809 + 0.587i)7-s + (−0.368 + 0.929i)9-s + (−0.860 + 0.509i)11-s + (−0.917 + 0.397i)13-s + (−0.481 − 0.876i)17-s + (−0.562 + 0.827i)19-s + (0.940 + 0.338i)21-s + (0.0627 + 0.998i)23-s + (0.975 − 0.218i)27-s + (−0.790 − 0.612i)29-s + (0.876 − 0.481i)31-s + (0.904 + 0.425i)33-s + (−0.218 + 0.975i)37-s + (0.844 + 0.535i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.455 + 0.890i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.455 + 0.890i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.455 + 0.890i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (1037, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ 0.455 + 0.890i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3155630112 + 0.1930365123i$
$L(\frac12,\chi)$  $\approx$  $0.3155630112 + 0.1930365123i$
$L(\chi,1)$  $\approx$  0.5959289214 - 0.07949141828i
$L(1,\chi)$  $\approx$  0.5959289214 - 0.07949141828i

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

−17.89068569223378256599657654966, −17.4897886423230145513494485016, −16.744123517088713381737875781917, −16.17550838529497859435548262982, −15.63622247356589668747229852763, −14.88491582275110370133535926656, −14.286269320131983643379488714605, −13.17082645705523343375473146175, −12.792460880490402902556756094901, −12.07068114220401463866191988193, −10.86424035668489478178002888221, −10.73712974853758704421198319604, −10.07974150054283890583908808964, −9.23111436286541453494692828982, −8.6467348621298951052319650319, −7.61710029237036813280945449215, −6.852409971635743261859432404362, −6.08917659718011594996626046777, −5.4853772570731665355636357983, −4.50924923787679534706191829318, −4.11254823362680332016401142037, −2.99919083981387921796534934646, −2.5555686420745224764334525755, −0.95430415616880573685872228247, −0.14001282347134699169829519762, 0.42172916066916476712714320757, 1.81624920094339335540594178801, 2.33611019424681665972840871220, 3.08678264089428477125319491806, 4.32207587422794087498689915429, 5.12943185871743307887476632268, 5.752935132143476028657751580393, 6.4909723200145556053001109346, 7.20369069063518611784513303757, 7.7570204154009903984194481554, 8.59643488313823022090062814801, 9.631801262065008689174342176507, 9.98340085407623738039802551582, 11.00150442961714278896620543417, 11.8142487569769311966862225416, 12.15682941136830693874821835599, 13.09310745197649569403064453827, 13.280174439499658710529910550659, 14.27538991402121957626164021969, 15.11928522996554029045706386316, 15.804403945160427949616909986753, 16.4299983493956290766421103819, 17.20997293971714578799920903907, 17.731122709962893811991694653312, 18.52835240542867529103452171858

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