Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.641 + 0.767i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.661 − 0.750i)3-s + (0.309 + 0.951i)7-s + (−0.125 − 0.992i)9-s + (0.940 − 0.338i)11-s + (−0.790 − 0.612i)13-s + (−0.770 − 0.637i)17-s + (0.661 + 0.750i)19-s + (0.917 + 0.397i)21-s + (0.876 − 0.481i)23-s + (−0.827 − 0.562i)27-s + (−0.218 − 0.975i)29-s + (−0.637 + 0.770i)31-s + (0.368 − 0.929i)33-s + (0.562 + 0.827i)37-s + (−0.982 + 0.187i)39-s + ⋯
L(s,χ)  = 1  + (0.661 − 0.750i)3-s + (0.309 + 0.951i)7-s + (−0.125 − 0.992i)9-s + (0.940 − 0.338i)11-s + (−0.790 − 0.612i)13-s + (−0.770 − 0.637i)17-s + (0.661 + 0.750i)19-s + (0.917 + 0.397i)21-s + (0.876 − 0.481i)23-s + (−0.827 − 0.562i)27-s + (−0.218 − 0.975i)29-s + (−0.637 + 0.770i)31-s + (0.368 − 0.929i)33-s + (0.562 + 0.827i)37-s + (−0.982 + 0.187i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.641 + 0.767i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.641 + 0.767i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.641 + 0.767i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (1253, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ 0.641 + 0.767i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.049153691 + 0.9583677370i$
$L(\frac12,\chi)$  $\approx$  $2.049153691 + 0.9583677370i$
$L(\chi,1)$  $\approx$  1.297565114 - 0.1828460267i
$L(1,\chi)$  $\approx$  1.297565114 - 0.1828460267i

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

−18.11009203949636103204196721449, −17.38222019767042119569876238244, −16.74493338217952324577862145356, −16.367737957885729869535544952071, −15.21749924091874940708655475625, −14.8985021444173094407745132042, −14.20905704484829641215696454831, −13.5914284719146068100842815930, −12.94895491792872256173479203763, −11.93673676267888212824858084254, −11.00957498936963447206543924592, −10.82758492044032145494004144774, −9.55446534116540121910430977398, −9.44680267786078047384255420725, −8.6298969803206603105496101182, −7.6197123599237125977339293903, −7.18220983799168351303635510671, −6.37790188827925526512005287764, −5.05754017424314025730544196577, −4.67917069096623394202018714218, −3.85534855008657684422585222830, −3.33280270678501007335577163832, −2.16709194460590162427806527299, −1.56101544897688984594082284082, −0.30990946705391421829238192552, 0.89392243542791076294896515984, 1.59531470585473503259884226590, 2.64119116558487711578488753052, 2.93993822232586628500243713854, 4.04544099073244893387374763886, 4.95256751933594804902747243002, 5.83868748156645058489379547411, 6.45717506890910708457446652199, 7.31837874458336096006621995148, 7.90868163473203277943077626760, 8.71694383135116047392957520810, 9.224030903997274370946882613167, 9.83100800091181425278318191968, 11.05358078398295092985333199591, 11.69596955763862746278295927112, 12.302218945724623089760381837512, 12.85360775002257390959673954339, 13.701027317841380380097736886404, 14.3954056440418624403179995385, 14.83827373164694720828826582137, 15.51355099651401043511035105652, 16.3376261051195574089387779692, 17.32938236756765979543963336423, 17.73375016760790946419211599544, 18.62179987597519682565886616118

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