Properties

Degree 1
Conductor $ 2^{3} \cdot 751 $
Sign $0.994 - 0.105i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.929 + 0.368i)3-s + (−0.535 − 0.844i)5-s + (−0.929 − 0.368i)7-s + (0.728 + 0.684i)9-s + (0.809 − 0.587i)11-s + (0.992 − 0.125i)13-s + (−0.187 − 0.982i)15-s + (0.0627 + 0.998i)17-s + (−0.876 − 0.481i)19-s + (−0.728 − 0.684i)21-s + (0.876 − 0.481i)23-s + (−0.425 + 0.904i)25-s + (0.425 + 0.904i)27-s + (−0.728 − 0.684i)29-s + (0.728 + 0.684i)31-s + ⋯
L(s,χ)  = 1  + (0.929 + 0.368i)3-s + (−0.535 − 0.844i)5-s + (−0.929 − 0.368i)7-s + (0.728 + 0.684i)9-s + (0.809 − 0.587i)11-s + (0.992 − 0.125i)13-s + (−0.187 − 0.982i)15-s + (0.0627 + 0.998i)17-s + (−0.876 − 0.481i)19-s + (−0.728 − 0.684i)21-s + (0.876 − 0.481i)23-s + (−0.425 + 0.904i)25-s + (0.425 + 0.904i)27-s + (−0.728 − 0.684i)29-s + (0.728 + 0.684i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.994 - 0.105i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.994 - 0.105i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $0.994 - 0.105i$
motivic weight  =  \(0\)
character  :  $\chi_{6008} (3917, \cdot )$
Sato-Tate  :  $\mu(50)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6008,\ (0:\ ),\ 0.994 - 0.105i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.364638957 - 0.1254246887i$
$L(\frac12,\chi)$  $\approx$  $2.364638957 - 0.1254246887i$
$L(\chi,1)$  $\approx$  1.359486030 - 0.05889468143i
$L(1,\chi)$  $\approx$  1.359486030 - 0.05889468143i

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

−18.047287370425667643323758781246, −17.007765346651547639810779526366, −16.32003432270302628378452046711, −15.46105747606385635878037213998, −15.12852543125445555693410253600, −14.60005910090853993026613806211, −13.62769614474672831220853663459, −13.38947592294328112686658284978, −12.43595826720664059535564370303, −11.881574173691444586603051529545, −11.24508231012990524616682230515, −10.1996847474021987973436553088, −9.77839516039181431759722717246, −8.87027479436917925886301111417, −8.55891909623016072076425090917, −7.51840844814329917336509832540, −6.89350492927047881562169998753, −6.60333743951877863280705320943, −5.71921034430567869607537035602, −4.49144981855115871535683622067, −3.600230370261420250037782798081, −3.436990149023501578541975724568, −2.48662480354815770221198421829, −1.837726684549533206939920442348, −0.71722841241656279570377493591, 0.79010627096484769337099992275, 1.463859826500663137025148771960, 2.59344195363392426877000531096, 3.4225923622700536683519276772, 3.960409503184641445616678427609, 4.362257210258451520433125939527, 5.423755821540018176350358137023, 6.35772891322312564828626090030, 6.86655738263389218801785529607, 7.99280129154726185466937746262, 8.34774269449272692917545574528, 9.10344464951084768483066060004, 9.38962876219479105610703399159, 10.47224353016741960689879954186, 10.906842175864896353400272583732, 11.80256262091884069724458470995, 12.78142374155559928363790736383, 13.08673975389858629617413584101, 13.624152198871493349646496484488, 14.50832539570370147401982807955, 15.191555794300552823411298813425, 15.683484581613753556560816491107, 16.482858377049281503687925246539, 16.71114488005845782579484470446, 17.514767451121818398744621078664

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