Properties

Degree 1
Conductor $ 2^{2} \cdot 751 $
Sign $0.811 + 0.584i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + 11-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)15-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.309 + 0.951i)23-s + (0.309 − 0.951i)25-s + (0.309 + 0.951i)27-s + (−0.309 + 0.951i)29-s + (0.309 − 0.951i)31-s + ⋯
L(s,χ)  = 1  + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + 11-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)15-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.309 + 0.951i)23-s + (0.309 − 0.951i)25-s + (0.309 + 0.951i)27-s + (−0.309 + 0.951i)29-s + (0.309 − 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 3004 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.811 + 0.584i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 3004 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.811 + 0.584i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(3004\)    =    \(2^{2} \cdot 751\)
\( \varepsilon \)  =  $0.811 + 0.584i$
motivic weight  =  \(0\)
character  :  $\chi_{3004} (359, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 3004,\ (0:\ ),\ 0.811 + 0.584i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8894914746 + 0.2868981250i$
$L(\frac12,\chi)$  $\approx$  $0.8894914746 + 0.2868981250i$
$L(\chi,1)$  $\approx$  0.6937514480 + 0.2293184503i
$L(1,\chi)$  $\approx$  0.6937514480 + 0.2293184503i

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

−18.99708322938299871864367963000, −18.45351312287337736997055513386, −17.329914126034781520598262309842, −16.95847499169205829921511828837, −16.26427610232032550331821752421, −15.84371702348110101232643950464, −14.80332984828787262291041768296, −13.926399009912598295784225734895, −13.1776914799835752026254096439, −12.51105191281049277002075651782, −12.02766525364184939371571571863, −11.43366598722957961571841773379, −10.36537818774907371751692461762, −10.02618748433437328937497120315, −8.80736637952100183777719492672, −8.095786483789795346554712237049, −7.38997090944307390204707356285, −6.69627591574432803399917904221, −5.9504722819391260811020263444, −5.21777239935833385596941958744, −4.182906497766343121599541505594, −3.6925720535463804803772158042, −2.58557356085121207523315658735, −1.14005259866457793817740555364, −0.7939551554243818428588192749, 0.52105800838200178718191707918, 1.805513808411222852105804380, 3.04500399944254506372851574467, 3.82031780211840034870822357187, 4.17996090044957484380514576086, 5.3673555594759948789437910973, 6.11698105164450706311995692249, 6.721563004583644934027331113604, 7.32695245694288757391196234375, 8.579340586302976358937334145740, 9.25118830693584930888270087483, 9.84149350099266347445291099175, 10.762377365046885829321410030845, 11.40614450502163126301066574103, 11.9265127394235668315598420627, 12.482840774390330592076321612072, 13.47643162543725825904146990513, 14.52840145218262993417188807373, 15.04103104666185906186510097914, 15.68566827926953577252595449069, 16.37369837205508512922012684044, 16.82109784171063741184481817064, 17.73152797032414647100202087121, 18.46088272840455003609343935620, 19.25120652994209923342629955405

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