Properties

Degree 1
Conductor $ 2^{3} \cdot 751 $
Sign $0.460 - 0.887i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.728 − 0.684i)3-s + (0.425 − 0.904i)5-s + (0.728 + 0.684i)7-s + (0.0627 + 0.998i)9-s + (−0.309 + 0.951i)11-s + (−0.968 + 0.248i)13-s + (−0.929 + 0.368i)15-s + (−0.992 + 0.125i)17-s + (−0.535 − 0.844i)19-s + (−0.0627 − 0.998i)21-s + (0.535 − 0.844i)23-s + (−0.637 − 0.770i)25-s + (0.637 − 0.770i)27-s + (−0.0627 − 0.998i)29-s + (0.0627 + 0.998i)31-s + ⋯
L(s,χ)  = 1  + (−0.728 − 0.684i)3-s + (0.425 − 0.904i)5-s + (0.728 + 0.684i)7-s + (0.0627 + 0.998i)9-s + (−0.309 + 0.951i)11-s + (−0.968 + 0.248i)13-s + (−0.929 + 0.368i)15-s + (−0.992 + 0.125i)17-s + (−0.535 − 0.844i)19-s + (−0.0627 − 0.998i)21-s + (0.535 − 0.844i)23-s + (−0.637 − 0.770i)25-s + (0.637 − 0.770i)27-s + (−0.0627 − 0.998i)29-s + (0.0627 + 0.998i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.460 - 0.887i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.460 - 0.887i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $0.460 - 0.887i$
motivic weight  =  \(0\)
character  :  $\chi_{6008} (1461, \cdot )$
Sato-Tate  :  $\mu(50)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6008,\ (0:\ ),\ 0.460 - 0.887i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.012680810 - 0.6151140473i$
$L(\frac12,\chi)$  $\approx$  $1.012680810 - 0.6151140473i$
$L(\chi,1)$  $\approx$  0.8381653383 - 0.2134447883i
$L(1,\chi)$  $\approx$  0.8381653383 - 0.2134447883i

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

−17.77470390693947007093763789520, −17.12264658822452683147919759298, −16.706848913907237485564772505318, −15.9063597323545848200641746460, −15.032669440042650205369865077227, −14.75389508395427370984908915223, −13.9983608952945572270572759459, −13.31357194373507060711484736437, −12.54739460845735339201054017609, −11.46577501764457123937621245549, −11.21274459805888823179927841476, −10.59308778181708344978287219605, −10.0825706350759669348626786340, −9.36366125689344590467450649361, −8.54154310096121297446366908001, −7.56499592593019898321248324674, −7.07642842642000730295904662294, −6.198876952908969195210067956589, −5.63093048652837829360145284857, −4.94927602794141227274290165754, −4.15361777996682906070361293112, −3.4860872669782134924251143489, −2.64995300917431665183497816391, −1.75065258945707889482014756553, −0.642967996268106162079703900098, 0.48431776407032699743961980739, 1.52508665942945561878143425845, 2.23073574019340014114963138252, 2.54871514944706964511479358504, 4.46255855522188712125962315925, 4.742822718071942307902315195532, 5.15938111083580301070254149053, 6.20003086005631274533116089841, 6.6464272307688632228057265087, 7.58351464980345198669480466970, 8.13891730667283525293395427789, 8.94739846466020573032847715064, 9.49965627635609843837524104866, 10.433817201114686378266326330908, 11.09078024773398135048659418210, 11.83826618987268506484851462812, 12.38919984349847158274139600334, 12.84613377731526049463582053088, 13.43339069783584134417380641470, 14.29173016441483241224390100696, 15.04048155550399403025260646592, 15.666392791555384384993573467009, 16.4515718144579519920095357181, 17.228634859414702593257991826523, 17.57280739052929796608798934673

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