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Properties

Label 1-6008-6008.4677-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.640 - 0.767i$
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  + (0.637 − 0.770i)3-s + (−0.968 + 0.248i)5-s + (−0.637 + 0.770i)7-s + (−0.187 − 0.982i)9-s + (0.809 − 0.587i)11-s + (−0.728 + 0.684i)13-s + (−0.425 + 0.904i)15-s + (−0.929 + 0.368i)17-s + (0.992 − 0.125i)19-s + (0.187 + 0.982i)21-s + (−0.992 − 0.125i)23-s + (0.876 − 0.481i)25-s + (−0.876 − 0.481i)27-s + (0.187 + 0.982i)29-s + (−0.187 − 0.982i)31-s + ⋯
L(s)  = 1  + (0.637 − 0.770i)3-s + (−0.968 + 0.248i)5-s + (−0.637 + 0.770i)7-s + (−0.187 − 0.982i)9-s + (0.809 − 0.587i)11-s + (−0.728 + 0.684i)13-s + (−0.425 + 0.904i)15-s + (−0.929 + 0.368i)17-s + (0.992 − 0.125i)19-s + (0.187 + 0.982i)21-s + (−0.992 − 0.125i)23-s + (0.876 − 0.481i)25-s + (−0.876 − 0.481i)27-s + (0.187 + 0.982i)29-s + (−0.187 − 0.982i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.640 - 0.767i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (4677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.640 - 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075329856 - 0.5030389232i\)
\(L(\frac12)\) \(\approx\) \(1.075329856 - 0.5030389232i\)
\(L(1)\) \(\approx\) \(0.9178204276 - 0.1575176521i\)
\(L(1)\) \(\approx\) \(0.9178204276 - 0.1575176521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.637 - 0.770i)T \)
5 \( 1 + (-0.968 + 0.248i)T \)
7 \( 1 + (-0.637 + 0.770i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.728 + 0.684i)T \)
17 \( 1 + (-0.929 + 0.368i)T \)
19 \( 1 + (0.992 - 0.125i)T \)
23 \( 1 + (-0.992 - 0.125i)T \)
29 \( 1 + (0.187 + 0.982i)T \)
31 \( 1 + (-0.187 - 0.982i)T \)
37 \( 1 + (-0.876 + 0.481i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.968 - 0.248i)T \)
47 \( 1 + (0.968 - 0.248i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.535 - 0.844i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.187 + 0.982i)T \)
71 \( 1 + (-0.425 - 0.904i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.968 - 0.248i)T \)
97 \( 1 + (-0.637 - 0.770i)T \)
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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

−17.6028285747000397808648049715, −17.05012713946983401156662520962, −16.32892316060403705241529085969, −15.70581830062640023742068709287, −15.42516910455297935560564106453, −14.55451092448916817996695196371, −13.96282102282891904271475245490, −13.35189755190952118417221450136, −12.468628465810330983347680912082, −11.888917260723971188930078636304, −11.18933009168801900639571503140, −10.2153466051653735242080868195, −9.98902031004806550773270493543, −9.11769686817757450817650269866, −8.57471299060629881426035511566, −7.619687623717669219018999040759, −7.319631986910254610261070975199, −6.498133058988741243656220998187, −5.27862380547450614289445366453, −4.71169446506568301965279453628, −3.92496236030886882977579189765, −3.5913916410652118200582062250, −2.76801729199124004200436575746, −1.81698494129220951122101780733, −0.59271241434101131266413436884, 0.43821638119936845673354548991, 1.60990194079959138965733751458, 2.35461812066347699945745157589, 3.15724681440052962195924514428, 3.67011116725003081865892939867, 4.44909270834976179881027451287, 5.54450055968077237580346708427, 6.417511386450185731420692121459, 6.86435827047408844461994259311, 7.45642609851147924110241124478, 8.43708054536286323732247935908, 8.731501208275120041177724069289, 9.45019660290248821522691142892, 10.20517397902574949427994562762, 11.43302331128103148424311083464, 11.746989378260032789864947307746, 12.27159130918066715203190004311, 12.95951641336376713251815019042, 13.82612842580566212356985764855, 14.24093299277958780780398549179, 15.14705431806729590491347145153, 15.399350751030915501593578840859, 16.31811765905680256077816798838, 16.85378879604891548628012636784, 17.86291627913592718972567789761

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