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Properties

Label 1-3004-3004.2435-r0-0-0
Degree $1$
Conductor $3004$
Sign $0.811 - 0.584i$
Analytic cond. $13.9505$
Root an. cond. $13.9505$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(3004\)    =    \(2^{2} \cdot 751\)
Sign: $0.811 - 0.584i$
Analytic conductor: \(13.9505\)
Root analytic conductor: \(13.9505\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3004} (2435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3004,\ (0:\ ),\ 0.811 - 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8894914746 - 0.2868981250i\)
\(L(\frac12)\) \(\approx\) \(0.8894914746 - 0.2868981250i\)
\(L(1)\) \(\approx\) \(0.6937514480 - 0.2293184503i\)
\(L(1)\) \(\approx\) \(0.6937514480 - 0.2293184503i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 - T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 - T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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