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Properties

Label 1-4004-4004.3139-r1-0-0
Degree $1$
Conductor $4004$
Sign $-0.302 - 0.953i$
Analytic cond. $430.289$
Root an. cond. $430.289$
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  + (0.913 − 0.406i)3-s + (−0.743 + 0.669i)5-s + (0.669 − 0.743i)9-s + (−0.406 + 0.913i)15-s + (0.309 − 0.951i)17-s + (0.994 + 0.104i)19-s + 23-s + (0.104 − 0.994i)25-s + (0.309 − 0.951i)27-s + (0.913 + 0.406i)29-s + (−0.743 − 0.669i)31-s + (−0.587 − 0.809i)37-s + (−0.994 − 0.104i)41-s + (−0.5 + 0.866i)43-s + i·45-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.743 + 0.669i)5-s + (0.669 − 0.743i)9-s + (−0.406 + 0.913i)15-s + (0.309 − 0.951i)17-s + (0.994 + 0.104i)19-s + 23-s + (0.104 − 0.994i)25-s + (0.309 − 0.951i)27-s + (0.913 + 0.406i)29-s + (−0.743 − 0.669i)31-s + (−0.587 − 0.809i)37-s + (−0.994 − 0.104i)41-s + (−0.5 + 0.866i)43-s + i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.302 - 0.953i$
Analytic conductor: \(430.289\)
Root analytic conductor: \(430.289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (3139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (1:\ ),\ -0.302 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.442332066 - 1.970263786i\)
\(L(\frac12)\) \(\approx\) \(1.442332066 - 1.970263786i\)
\(L(1)\) \(\approx\) \(1.296432854 - 0.2473476721i\)
\(L(1)\) \(\approx\) \(1.296432854 - 0.2473476721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (-0.743 + 0.669i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.994 + 0.104i)T \)
23 \( 1 + T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (-0.743 - 0.669i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.994 - 0.104i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.406 + 0.913i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.743 + 0.669i)T \)
73 \( 1 + (0.406 - 0.913i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.207 - 0.978i)T \)
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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

−18.796330361731803112139968862412, −17.88200971699660581772274279237, −16.83127000233802061896378885031, −16.53404912224222396146818012977, −15.53303358445068724699475818796, −15.30646768658513379569705303572, −14.54926769982033455986670818494, −13.6468660275218368432384697072, −13.23004727998116900248876783058, −12.29260432481801116044909358993, −11.82371201091763332230291068004, −10.77384072419842558939807174517, −10.20483614565407486372394771584, −9.36620650425847420289610764952, −8.617336516473214357517517121965, −8.32318219172570734738409553965, −7.39864752846238285465122738345, −6.871396598041163077404929428916, −5.49662846496483658461281479718, −4.9587126345348372447889018285, −4.117912258718161853608356665574, −3.48308878681825123271826064147, −2.83779507411164078461340495024, −1.66440511882451070385816150392, −0.98525105871111888146967381299, 0.3418643581996730043432430534, 1.21344735835417744044850504694, 2.26085522340884005564106423164, 3.134722465611963510557301189, 3.399631798195667414384894546590, 4.43078384731614836649208466705, 5.26284460749710664327678175574, 6.39872153592845306079093297387, 7.132054523564942113130151427743, 7.49990680529274871654775325018, 8.251890130281571745078203348017, 9.02484175611996651545931175430, 9.69188349355454519200676058333, 10.46902030642901576633507658953, 11.36376681279712280172122854676, 11.92045772238947182441745480363, 12.65904938847529285361666811789, 13.42716831749583760993564728330, 14.190712685125772187023252455368, 14.546785112371355727948673541683, 15.36532844409094869087981733573, 15.88174750022427951522436613994, 16.588449140441269227813026385147, 17.73957697289379336826617106363, 18.276348561227751671328858299234

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