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Properties

Label 1-6008-6008.3485-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.959 + 0.281i$
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  + (−0.968 − 0.248i)3-s + (0.929 − 0.368i)5-s + (0.968 + 0.248i)7-s + (0.876 + 0.481i)9-s + (0.809 + 0.587i)11-s + (0.425 − 0.904i)13-s + (−0.992 + 0.125i)15-s + (0.535 + 0.844i)17-s + (0.187 + 0.982i)19-s + (−0.876 − 0.481i)21-s + (−0.187 + 0.982i)23-s + (0.728 − 0.684i)25-s + (−0.728 − 0.684i)27-s + (−0.876 − 0.481i)29-s + (0.876 + 0.481i)31-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)3-s + (0.929 − 0.368i)5-s + (0.968 + 0.248i)7-s + (0.876 + 0.481i)9-s + (0.809 + 0.587i)11-s + (0.425 − 0.904i)13-s + (−0.992 + 0.125i)15-s + (0.535 + 0.844i)17-s + (0.187 + 0.982i)19-s + (−0.876 − 0.481i)21-s + (−0.187 + 0.982i)23-s + (0.728 − 0.684i)25-s + (−0.728 − 0.684i)27-s + (−0.876 − 0.481i)29-s + (0.876 + 0.481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.959 + 0.281i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (3485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.959 + 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.174274543 + 0.3121787598i\)
\(L(\frac12)\) \(\approx\) \(2.174274543 + 0.3121787598i\)
\(L(1)\) \(\approx\) \(1.211356005 + 0.0002029165575i\)
\(L(1)\) \(\approx\) \(1.211356005 + 0.0002029165575i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.968 - 0.248i)T \)
5 \( 1 + (0.929 - 0.368i)T \)
7 \( 1 + (0.968 + 0.248i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.425 - 0.904i)T \)
17 \( 1 + (0.535 + 0.844i)T \)
19 \( 1 + (0.187 + 0.982i)T \)
23 \( 1 + (-0.187 + 0.982i)T \)
29 \( 1 + (-0.876 - 0.481i)T \)
31 \( 1 + (0.876 + 0.481i)T \)
37 \( 1 + (-0.728 + 0.684i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.929 + 0.368i)T \)
47 \( 1 + (-0.929 + 0.368i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.0627 - 0.998i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.876 - 0.481i)T \)
71 \( 1 + (-0.992 - 0.125i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.187 + 0.982i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.929 + 0.368i)T \)
97 \( 1 + (0.968 - 0.248i)T \)
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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

−17.68759751688187234236363360512, −16.910312244076650510968007565391, −16.66764988547220236350933035384, −15.90980156322107786311183420003, −14.932860450368445476853700692045, −14.416308248359859439138625105306, −13.745821354968925360442472464521, −13.27383951178484880375538950482, −12.14768835753209622026559920484, −11.61183614601762643205488270342, −11.12441282167795154352907245931, −10.52727556789774065656229090318, −9.79766504109618237665658874151, −9.088976763478432986634790830715, −8.54405235500232925664151351563, −7.15207791438514738558056595925, −6.99657452069829630070972637323, −6.04283561823860471111513596631, −5.57624057088061611153319786911, −4.742256239474087508649082861528, −4.218236986327704821764667838698, −3.259469279521653232530089267311, −2.22024772366789698793196790211, −1.41649043614747993677820350031, −0.75577887912015966338688682978, 0.98141556360666246188460053447, 1.61551802244752670874010798354, 1.960193875814770826621748527190, 3.37959469372203060641342631850, 4.21448208457265979283091953355, 5.05946576802917819161634094226, 5.55327432754936892040656619944, 6.06565964309810492979421227631, 6.79370958036537445179016461361, 7.780713808313672006552903432477, 8.22048535993891180927744264040, 9.15975782373029147792772726646, 10.063180794703949575052124568026, 10.28954566460881159225452032091, 11.245371546207174185915601853622, 11.851437096951471999704229167914, 12.43644589695382219851093317563, 13.00429640604560566745457323775, 13.78406791434997916720523748120, 14.41245850996829749430735854029, 15.16509233220236677093178161652, 15.82147744707568111989834795889, 16.74592173815066302526324348403, 17.27443913748077577563574979451, 17.55964280916722625739042676982

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