Properties

Label 1-6008-6008.195-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.765 + 0.643i$
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  + (−0.535 − 0.844i)3-s + (−0.0627 + 0.998i)5-s + (0.535 + 0.844i)7-s + (−0.425 + 0.904i)9-s + (0.809 − 0.587i)11-s + (0.187 − 0.982i)13-s + (0.876 − 0.481i)15-s + (0.637 + 0.770i)17-s + (0.728 − 0.684i)19-s + (0.425 − 0.904i)21-s + (−0.728 − 0.684i)23-s + (−0.992 − 0.125i)25-s + (0.992 − 0.125i)27-s + (−0.425 + 0.904i)29-s + (−0.425 + 0.904i)31-s + ⋯
L(s)  = 1  + (−0.535 − 0.844i)3-s + (−0.0627 + 0.998i)5-s + (0.535 + 0.844i)7-s + (−0.425 + 0.904i)9-s + (0.809 − 0.587i)11-s + (0.187 − 0.982i)13-s + (0.876 − 0.481i)15-s + (0.637 + 0.770i)17-s + (0.728 − 0.684i)19-s + (0.425 − 0.904i)21-s + (−0.728 − 0.684i)23-s + (−0.992 − 0.125i)25-s + (0.992 − 0.125i)27-s + (−0.425 + 0.904i)29-s + (−0.425 + 0.904i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.765 + 0.643i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.573088352 + 0.5729812879i\)
\(L(\frac12)\) \(\approx\) \(1.573088352 + 0.5729812879i\)
\(L(1)\) \(\approx\) \(1.032621432 + 0.04379094869i\)
\(L(1)\) \(\approx\) \(1.032621432 + 0.04379094869i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.535 - 0.844i)T \)
5 \( 1 + (-0.0627 + 0.998i)T \)
7 \( 1 + (0.535 + 0.844i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.187 - 0.982i)T \)
17 \( 1 + (0.637 + 0.770i)T \)
19 \( 1 + (0.728 - 0.684i)T \)
23 \( 1 + (-0.728 - 0.684i)T \)
29 \( 1 + (-0.425 + 0.904i)T \)
31 \( 1 + (-0.425 + 0.904i)T \)
37 \( 1 + (0.992 + 0.125i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (0.0627 + 0.998i)T \)
47 \( 1 + (-0.0627 + 0.998i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.968 - 0.248i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.425 - 0.904i)T \)
71 \( 1 + (-0.876 - 0.481i)T \)
73 \( 1 - T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.0627 - 0.998i)T \)
97 \( 1 + (0.535 - 0.844i)T \)
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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

−17.4327739998870932208077545445, −16.90007132459923002988037484143, −16.27371851854495750563774004042, −16.10324799390893007719764621440, −14.93444728871668719584656815662, −14.49223319980114293259679046548, −13.749478524117331787794191101237, −13.10881392864457562852898817187, −11.89719219771423889666947934825, −11.82362166848474034915348798977, −11.26209240931083855775486815330, −10.10773098134295037182379939133, −9.681339727503983077261439854929, −9.2489361695017829603402527844, −8.32815503952695637254272749702, −7.56813386671748720229886586130, −6.90788089839066163795109213228, −5.84154071337624362122661815443, −5.40599052062551525187985271019, −4.51286409711020176376806845195, −4.02759437911856828687207312764, −3.674656249734569782833506051085, −2.14523434011535551901154915217, −1.31337895763061866347103314883, −0.58562540252911287127366018589, 0.893290856296486345086299052270, 1.60496187843371406855263688269, 2.577601176991994334403907468622, 3.06655821279377566437649439559, 4.0144109506480222847025356061, 5.13284246703460904123464396063, 5.81979963841945131659903130214, 6.15965790453837592216086858560, 6.99726667009658110298175125104, 7.699429871611599346746975542581, 8.25238189040583470283667272486, 8.97686039195905924158846257000, 9.95476982028522580747094066885, 10.88663686028669876652169079155, 11.072911898688449560802122451662, 11.89207733065660383868537086064, 12.405341222780839156133478474838, 13.09111171536498004585295128651, 13.97433224795637288677380109131, 14.526171064036913718172371802032, 14.92937562653726470802639792669, 15.96827217159522006559767889706, 16.417145748793565565250653527833, 17.53620692805554693154357194732, 17.765115140628658583654529010402

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