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Properties

Label 1-6008-6008.925-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.692 - 0.721i$
Analytic cond. $27.9010$
Root an. cond. $27.9010$
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  + (0.455 − 0.890i)3-s + (0.941 − 0.336i)5-s + (0.260 − 0.965i)7-s + (−0.584 − 0.811i)9-s + (0.756 + 0.653i)11-s + (0.121 − 0.992i)13-s + (0.129 − 0.991i)15-s + (0.976 + 0.216i)17-s + (−0.372 − 0.928i)19-s + (−0.740 − 0.672i)21-s + (0.884 + 0.467i)23-s + (0.773 − 0.634i)25-s + (−0.988 + 0.150i)27-s + (0.203 + 0.979i)29-s + (−0.952 + 0.305i)31-s + ⋯
L(s)  = 1  + (0.455 − 0.890i)3-s + (0.941 − 0.336i)5-s + (0.260 − 0.965i)7-s + (−0.584 − 0.811i)9-s + (0.756 + 0.653i)11-s + (0.121 − 0.992i)13-s + (0.129 − 0.991i)15-s + (0.976 + 0.216i)17-s + (−0.372 − 0.928i)19-s + (−0.740 − 0.672i)21-s + (0.884 + 0.467i)23-s + (0.773 − 0.634i)25-s + (−0.988 + 0.150i)27-s + (0.203 + 0.979i)29-s + (−0.952 + 0.305i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.692 - 0.721i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (925, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.692 - 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.164822164 - 2.733929403i\)
\(L(\frac12)\) \(\approx\) \(1.164822164 - 2.733929403i\)
\(L(1)\) \(\approx\) \(1.341762780 - 0.8715068028i\)
\(L(1)\) \(\approx\) \(1.341762780 - 0.8715068028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.455 - 0.890i)T \)
5 \( 1 + (0.941 - 0.336i)T \)
7 \( 1 + (0.260 - 0.965i)T \)
11 \( 1 + (0.756 + 0.653i)T \)
13 \( 1 + (0.121 - 0.992i)T \)
17 \( 1 + (0.976 + 0.216i)T \)
19 \( 1 + (-0.372 - 0.928i)T \)
23 \( 1 + (0.884 + 0.467i)T \)
29 \( 1 + (0.203 + 0.979i)T \)
31 \( 1 + (-0.952 + 0.305i)T \)
37 \( 1 + (0.998 - 0.0586i)T \)
41 \( 1 + (-0.187 + 0.982i)T \)
43 \( 1 + (-0.850 - 0.525i)T \)
47 \( 1 + (-0.611 - 0.791i)T \)
53 \( 1 + (0.992 + 0.125i)T \)
59 \( 1 + (-0.859 - 0.510i)T \)
61 \( 1 + (-0.387 - 0.921i)T \)
67 \( 1 + (0.994 + 0.108i)T \)
71 \( 1 + (-0.999 + 0.0251i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.372 - 0.928i)T \)
83 \( 1 + (-0.985 - 0.166i)T \)
89 \( 1 + (-0.880 - 0.474i)T \)
97 \( 1 + (0.705 - 0.708i)T \)
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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

−18.0499575781913110830868876416, −16.943915399452668354674103986049, −16.74448019332417056256007450393, −16.121459344161081143455553406, −15.0504909887700964960603012807, −14.718054572834189718861485641891, −14.19080695744813912425646148380, −13.6304989661770481213467581911, −12.762147687112775721657005977804, −11.83767440353813138409092619897, −11.32470662636920373477146397990, −10.607443708664968398800783726227, −9.81230372919255402541648273921, −9.34449925989007019605690026307, −8.79483067300965351697456213576, −8.1895125215344271681884602928, −7.20632251183653490475307096967, −6.124174531265267291874858808, −5.88461100686756632636452360185, −5.05232551773820031984650673305, −4.261694531188074949630247253074, −3.45339550425027117866067868586, −2.75445358610158188758719712588, −2.05352910534910494073143425922, −1.301313218680390494834402141434, 0.67821921086364272555645965660, 1.37426012090753836022983826005, 1.84544125808045463388528038916, 2.97181231164726461185995146739, 3.47201993795078991165340578913, 4.57275202320047713550465825206, 5.27568484671467192200289051057, 6.04055408117560654106399711266, 6.913670735070726669386147326456, 7.19461373284141279670893802815, 8.10711274304133084579351537710, 8.72816681006400303004273931141, 9.468032731401897741071906130850, 10.05592625319109788380253503937, 10.8307445809552737923678911804, 11.58969957749541054651181794382, 12.48900490914903420479692239726, 13.044021592998634557852064770300, 13.3332153188041609283294342858, 14.243028573074238551144505722138, 14.62432766726583704700122850443, 15.228186235447477477842307069879, 16.4560833508505196763759407302, 17.08886426072006904440922276748, 17.39489285119525766780417002880

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