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Properties

Label 1-6008-6008.421-r1-0-0
Degree $1$
Conductor $6008$
Sign $-0.259 - 0.965i$
Analytic cond. $645.649$
Root an. cond. $645.649$
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  + (−0.998 + 0.0586i)3-s + (−0.681 − 0.731i)5-s + (0.888 − 0.459i)7-s + (0.993 − 0.117i)9-s + (0.463 − 0.886i)11-s + (0.0544 − 0.998i)13-s + (0.723 + 0.690i)15-s + (0.903 − 0.429i)17-s + (0.999 − 0.00837i)19-s + (−0.859 + 0.510i)21-s + (0.976 + 0.216i)23-s + (−0.0711 + 0.997i)25-s + (−0.984 + 0.175i)27-s + (0.751 + 0.659i)29-s + (0.872 + 0.489i)31-s + ⋯
L(s)  = 1  + (−0.998 + 0.0586i)3-s + (−0.681 − 0.731i)5-s + (0.888 − 0.459i)7-s + (0.993 − 0.117i)9-s + (0.463 − 0.886i)11-s + (0.0544 − 0.998i)13-s + (0.723 + 0.690i)15-s + (0.903 − 0.429i)17-s + (0.999 − 0.00837i)19-s + (−0.859 + 0.510i)21-s + (0.976 + 0.216i)23-s + (−0.0711 + 0.997i)25-s + (−0.984 + 0.175i)27-s + (0.751 + 0.659i)29-s + (0.872 + 0.489i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.259 - 0.965i$
Analytic conductor: \(645.649\)
Root analytic conductor: \(645.649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (1:\ ),\ -0.259 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413136820 - 1.842537754i\)
\(L(\frac12)\) \(\approx\) \(1.413136820 - 1.842537754i\)
\(L(1)\) \(\approx\) \(0.9203422185 - 0.3426143366i\)
\(L(1)\) \(\approx\) \(0.9203422185 - 0.3426143366i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.998 + 0.0586i)T \)
5 \( 1 + (-0.681 - 0.731i)T \)
7 \( 1 + (0.888 - 0.459i)T \)
11 \( 1 + (0.463 - 0.886i)T \)
13 \( 1 + (0.0544 - 0.998i)T \)
17 \( 1 + (0.903 - 0.429i)T \)
19 \( 1 + (0.999 - 0.00837i)T \)
23 \( 1 + (0.976 + 0.216i)T \)
29 \( 1 + (0.751 + 0.659i)T \)
31 \( 1 + (0.872 + 0.489i)T \)
37 \( 1 + (0.970 - 0.240i)T \)
41 \( 1 + (-0.535 - 0.844i)T \)
43 \( 1 + (-0.920 + 0.391i)T \)
47 \( 1 + (-0.981 - 0.191i)T \)
53 \( 1 + (0.929 - 0.368i)T \)
59 \( 1 + (-0.244 - 0.969i)T \)
61 \( 1 + (-0.146 + 0.989i)T \)
67 \( 1 + (-0.220 - 0.975i)T \)
71 \( 1 + (0.850 - 0.525i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.999 + 0.00837i)T \)
83 \( 1 + (-0.0209 + 0.999i)T \)
89 \( 1 + (-0.799 + 0.601i)T \)
97 \( 1 + (0.842 - 0.539i)T \)
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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

−17.80332537026140359594127101684, −17.05021360406655619672299302076, −16.5932106317970590564195990726, −15.7176472027623470066495120441, −15.11186298504877192066677203110, −14.66585712063970511380543140178, −13.91552854125299011656469520010, −13.019435452309622983006630241843, −12.01294617646055977735948179962, −11.79822375249351596770095773673, −11.43316552739825499236983098032, −10.525496897149535450821455908645, −9.93508272345115265823632320761, −9.21555017419693172243973994268, −8.090950495162877934539190056650, −7.67530406132562390395495894109, −6.80335479139820014069965788746, −6.434803109097995536225144990, −5.4965501159299696747553488378, −4.66117172075331605852359751665, −4.33880003770271742279402680164, −3.34185757127518153664685389217, −2.36774227951239704092456064349, −1.48386276199481146383538484464, −0.82195250672138056682027763358, 0.58760244277829657373998954399, 0.88459069181927526558504234791, 1.48981396288220058156592743102, 3.10645375366213708877509341021, 3.6253960856379914563462757511, 4.5792542533913200381923687274, 5.222161302026949675844458961966, 5.454046717270511973386031396909, 6.59038320975792253440939616631, 7.28864265964541304783423749475, 7.99256928457002243041991921530, 8.47483052535861637108036328081, 9.45737016858610437817375576468, 10.17587796980430015312831981992, 10.96989633252312648366052187072, 11.41082425328834910662416864016, 11.99790658009293746756122894977, 12.54210509117378234040511714125, 13.39947665627890483722416026317, 13.9685480951645111375063893841, 14.91538590420093539253276956117, 15.51460194898610538508914179738, 16.25747016360600558257197767320, 16.70569695036537207104837761871, 17.18697384727871083580444289591

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