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Properties

Label 1-6008-6008.53-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.555 + 0.831i$
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  + (−0.876 − 0.481i)3-s + (−0.728 + 0.684i)5-s + (0.876 + 0.481i)7-s + (0.535 + 0.844i)9-s + (−0.309 − 0.951i)11-s + (0.637 + 0.770i)13-s + (0.968 − 0.248i)15-s + (−0.425 + 0.904i)17-s + (0.929 − 0.368i)19-s + (−0.535 − 0.844i)21-s + (−0.929 − 0.368i)23-s + (0.0627 − 0.998i)25-s + (−0.0627 − 0.998i)27-s + (−0.535 − 0.844i)29-s + (0.535 + 0.844i)31-s + ⋯
L(s)  = 1  + (−0.876 − 0.481i)3-s + (−0.728 + 0.684i)5-s + (0.876 + 0.481i)7-s + (0.535 + 0.844i)9-s + (−0.309 − 0.951i)11-s + (0.637 + 0.770i)13-s + (0.968 − 0.248i)15-s + (−0.425 + 0.904i)17-s + (0.929 − 0.368i)19-s + (−0.535 − 0.844i)21-s + (−0.929 − 0.368i)23-s + (0.0627 − 0.998i)25-s + (−0.0627 − 0.998i)27-s + (−0.535 − 0.844i)29-s + (0.535 + 0.844i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.006280206 + 0.5378585687i\)
\(L(\frac12)\) \(\approx\) \(1.006280206 + 0.5378585687i\)
\(L(1)\) \(\approx\) \(0.8022349729 + 0.08210632097i\)
\(L(1)\) \(\approx\) \(0.8022349729 + 0.08210632097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.876 - 0.481i)T \)
5 \( 1 + (-0.728 + 0.684i)T \)
7 \( 1 + (0.876 + 0.481i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.637 + 0.770i)T \)
17 \( 1 + (-0.425 + 0.904i)T \)
19 \( 1 + (0.929 - 0.368i)T \)
23 \( 1 + (-0.929 - 0.368i)T \)
29 \( 1 + (-0.535 - 0.844i)T \)
31 \( 1 + (0.535 + 0.844i)T \)
37 \( 1 + (-0.0627 + 0.998i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.728 - 0.684i)T \)
47 \( 1 + (0.728 - 0.684i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.992 - 0.125i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.535 - 0.844i)T \)
71 \( 1 + (0.968 + 0.248i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.929 - 0.368i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (0.728 - 0.684i)T \)
97 \( 1 + (0.876 - 0.481i)T \)
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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

−17.73266102208751998716624774507, −16.938700748207860835397707331239, −16.183985970305436993887232159131, −15.788327330843176702222899185739, −15.242148757245535451079124400, −14.458325545884957567207401459505, −13.62134410700237577516871270123, −12.80388131324563493468585366611, −12.2787483539144680181866742194, −11.51116331506971061445656286753, −11.19864577664447831297314261748, −10.3922644889421015177663227925, −9.71257095191254153657716630314, −9.04955849172191611624615274447, −8.07167100475982699552525262638, −7.5696142395934438984796083104, −6.98172535741430470027375778956, −5.84743128117220812844523942222, −5.189799156207487904490376764186, −4.78464440876531238262451349883, −3.97747666466126936774504198981, −3.508618350744094967745372811397, −2.12699671044883842941790891246, −1.18558592561354099029178884237, −0.49272626875365289867527762551, 0.75918425672920093399503064780, 1.67739177131272031200977691998, 2.42378530841233187203978086282, 3.39937096038060855963273270339, 4.23617131997641727305878503462, 4.87725945758188219539417376223, 5.8117168884177211147682231312, 6.26629487770668258504373656399, 6.96314801826386248118164377108, 7.84644019935689799207901666319, 8.22749126706337108793968271341, 8.95066108983680524697515061646, 10.20737858905116480932653033509, 10.705856832100652650066800761245, 11.42755476018029164879134710779, 11.685523479298102933395720640586, 12.252641055132050905627813006605, 13.34978393281312064306978256904, 13.78367882322117167483058032539, 14.50437003873108849592909274570, 15.43052455052992721202264106009, 15.78328824890795186021721656934, 16.52614554530953362399316310289, 17.20862836087425206847636110318, 17.9958701127649704116418932576

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