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Properties

Label 1-6008-6008.37-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.661 - 0.750i$
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  + (0.650 − 0.759i)3-s + (−0.884 + 0.467i)5-s + (0.823 + 0.567i)7-s + (−0.154 − 0.988i)9-s + (0.348 − 0.937i)11-s + (−0.0961 − 0.995i)13-s + (−0.220 + 0.975i)15-s + (−0.410 + 0.911i)17-s + (−0.340 − 0.940i)19-s + (0.966 − 0.256i)21-s + (0.926 − 0.375i)23-s + (0.563 − 0.825i)25-s + (−0.850 − 0.525i)27-s + (0.0544 − 0.998i)29-s + (0.705 + 0.708i)31-s + ⋯
L(s)  = 1  + (0.650 − 0.759i)3-s + (−0.884 + 0.467i)5-s + (0.823 + 0.567i)7-s + (−0.154 − 0.988i)9-s + (0.348 − 0.937i)11-s + (−0.0961 − 0.995i)13-s + (−0.220 + 0.975i)15-s + (−0.410 + 0.911i)17-s + (−0.340 − 0.940i)19-s + (0.966 − 0.256i)21-s + (0.926 − 0.375i)23-s + (0.563 − 0.825i)25-s + (−0.850 − 0.525i)27-s + (0.0544 − 0.998i)29-s + (0.705 + 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.661 - 0.750i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.661 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7176857224 - 1.589149001i\)
\(L(\frac12)\) \(\approx\) \(0.7176857224 - 1.589149001i\)
\(L(1)\) \(\approx\) \(1.110925164 - 0.4403450212i\)
\(L(1)\) \(\approx\) \(1.110925164 - 0.4403450212i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.650 - 0.759i)T \)
5 \( 1 + (-0.884 + 0.467i)T \)
7 \( 1 + (0.823 + 0.567i)T \)
11 \( 1 + (0.348 - 0.937i)T \)
13 \( 1 + (-0.0961 - 0.995i)T \)
17 \( 1 + (-0.410 + 0.911i)T \)
19 \( 1 + (-0.340 - 0.940i)T \)
23 \( 1 + (0.926 - 0.375i)T \)
29 \( 1 + (0.0544 - 0.998i)T \)
31 \( 1 + (0.705 + 0.708i)T \)
37 \( 1 + (0.611 + 0.791i)T \)
41 \( 1 + (-0.637 - 0.770i)T \)
43 \( 1 + (0.556 - 0.830i)T \)
47 \( 1 + (-0.989 - 0.141i)T \)
53 \( 1 + (-0.0627 + 0.998i)T \)
59 \( 1 + (-0.0460 - 0.998i)T \)
61 \( 1 + (-0.832 - 0.553i)T \)
67 \( 1 + (0.542 + 0.839i)T \)
71 \( 1 + (0.577 - 0.816i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.340 - 0.940i)T \)
83 \( 1 + (-0.996 - 0.0836i)T \)
89 \( 1 + (0.997 - 0.0669i)T \)
97 \( 1 + (0.0795 + 0.996i)T \)
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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

−17.94914887065364108015718704961, −16.99142600851874660228925120917, −16.63123552805701875583013640114, −16.00801832792159942910414123217, −15.24534724648199266730223235525, −14.653168744281607669927433999134, −14.31664730327737511262014744321, −13.437945731985365097569900385545, −12.74853438982029756466208410165, −11.8282207338045787371657217861, −11.33592639163045404757429045958, −10.78445388141059075950112829481, −9.75564786981042897470793169037, −9.38183417349961206812355884720, −8.56281095763631258458518165594, −8.017922122347469067733841080688, −7.30753146354308675779457888594, −6.8056938576183469386903932337, −5.418116425244655200478740766730, −4.58538021775270407952804742792, −4.4542243096044293749946536681, −3.739414287760816736412175138375, −2.83641174179181135295196821651, −1.865593170583567627163497965530, −1.18049332251608682821766727383, 0.42138603430208049499086684179, 1.24521252659451102618310190606, 2.29359196436902899545164605379, 2.896547143299405714581840873291, 3.51205534123986469364700511599, 4.39164220058843872658978254163, 5.20739901983924686520424007347, 6.282094414555829291920802650558, 6.60191643433719569776268060192, 7.65335915220453669441053589795, 8.054616403518449932802531320803, 8.65167028620302456422632408056, 9.088402823103449933343476295148, 10.39208075972438477528206405501, 10.95807107003766658147270531550, 11.62002694828212375798279424160, 12.167380830615941456135323736239, 12.896033145125707800315871979357, 13.55532161976423462976624627125, 14.25459751183710240117079492451, 15.024618236614946576696598440842, 15.22334237082112710823836196545, 15.86867508539959671948798480684, 17.17888213315084650352094089020, 17.43982106866050770195918088392

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