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Properties

Label 1-6008-6008.4685-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.813 - 0.581i$
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  + (−0.0627 − 0.998i)3-s + (0.637 + 0.770i)5-s + (0.0627 + 0.998i)7-s + (−0.992 + 0.125i)9-s + (0.809 + 0.587i)11-s + (−0.876 + 0.481i)13-s + (0.728 − 0.684i)15-s + (0.968 − 0.248i)17-s + (0.425 − 0.904i)19-s + (0.992 − 0.125i)21-s + (−0.425 − 0.904i)23-s + (−0.187 + 0.982i)25-s + (0.187 + 0.982i)27-s + (0.992 − 0.125i)29-s + (−0.992 + 0.125i)31-s + ⋯
L(s)  = 1  + (−0.0627 − 0.998i)3-s + (0.637 + 0.770i)5-s + (0.0627 + 0.998i)7-s + (−0.992 + 0.125i)9-s + (0.809 + 0.587i)11-s + (−0.876 + 0.481i)13-s + (0.728 − 0.684i)15-s + (0.968 − 0.248i)17-s + (0.425 − 0.904i)19-s + (0.992 − 0.125i)21-s + (−0.425 − 0.904i)23-s + (−0.187 + 0.982i)25-s + (0.187 + 0.982i)27-s + (0.992 − 0.125i)29-s + (−0.992 + 0.125i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.813 - 0.581i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (4685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.813 - 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.905881655 - 0.6113579293i\)
\(L(\frac12)\) \(\approx\) \(1.905881655 - 0.6113579293i\)
\(L(1)\) \(\approx\) \(1.190716263 - 0.1378978200i\)
\(L(1)\) \(\approx\) \(1.190716263 - 0.1378978200i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.0627 - 0.998i)T \)
5 \( 1 + (0.637 + 0.770i)T \)
7 \( 1 + (0.0627 + 0.998i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.876 + 0.481i)T \)
17 \( 1 + (0.968 - 0.248i)T \)
19 \( 1 + (0.425 - 0.904i)T \)
23 \( 1 + (-0.425 - 0.904i)T \)
29 \( 1 + (0.992 - 0.125i)T \)
31 \( 1 + (-0.992 + 0.125i)T \)
37 \( 1 + (0.187 - 0.982i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.637 - 0.770i)T \)
47 \( 1 + (-0.637 - 0.770i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.929 - 0.368i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.992 - 0.125i)T \)
71 \( 1 + (0.728 + 0.684i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.425 - 0.904i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.637 - 0.770i)T \)
97 \( 1 + (0.0627 - 0.998i)T \)
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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

−17.52326588720301253751783989184, −16.89101813080879598306043552112, −16.6108286516742794203623170085, −16.09440860710115336018739624508, −15.11205911001642087239790442339, −14.338421574626167715583488786403, −14.083922235637713558396821236798, −13.316552236447111350471618252259, −12.41296002653083885459313394724, −11.85275926562456231539466946507, −11.0732530797228144936869175141, −10.28223583318861327507536846077, −9.67164451108319416125802887094, −9.559028899638736231032809875770, −8.283058348627986230962337301772, −8.07311782377958572363001785445, −6.953823034798912058430254314181, −6.040576007866087972204190863009, −5.46368436166670801003096244041, −4.88098601776428002130659416156, −4.019615709313153625388336458287, −3.55101138960667642578248927609, −2.68172701764467650178090910213, −1.42145662953891259531408803679, −0.881663634313053019623427397139, 0.60555639642466914738628289623, 1.8622782553718215870827248962, 2.17287565701990430798007540965, 2.850352859455094379607711512143, 3.74293321185705031392422809832, 5.035000972339609848725980098677, 5.46712016392537159713430023083, 6.302445493520612058323439784163, 6.966785497983408754724049039143, 7.23240267512953912612237232539, 8.29647612673027598850368036100, 8.983860262308599095645297553024, 9.62926386334201205534614762110, 10.26539226175841479475991470865, 11.33943366468599406415779971236, 11.75176136077031565325334579644, 12.408514437345822879127743022793, 12.8781833109884514442528071390, 13.92873238829724407484157592348, 14.37377917650720367405893380405, 14.69494728589603244693363343836, 15.58621369889376604392225332798, 16.52837080450725661309110311446, 17.21595358082865230879271815747, 17.76912571236334173174672488562

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