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Properties

Label 1-6008-6008.4221-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.994 - 0.103i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.994 - 0.103i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (4221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.994 - 0.103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.516988487 - 0.07852527527i\)
\(L(\frac12)\) \(\approx\) \(1.516988487 - 0.07852527527i\)
\(L(1)\) \(\approx\) \(0.9723144028 + 0.02734363322i\)
\(L(1)\) \(\approx\) \(0.9723144028 + 0.02734363322i\)

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

−17.889871291747456087706392745526, −17.18529496579041287306583394448, −16.64056668151211076719836622418, −15.50009665354914753472164368806, −15.23656254940904190544616746941, −14.34815940806326161453722073420, −13.81267561484566359316045946808, −13.15948593191926252654149833614, −12.29990095403915297104683233825, −11.73754433156064100521921475500, −11.21614936891050489019151991588, −10.71786247958673080438750324687, −9.869970371409408258204403727101, −8.849148299390801969363733646472, −8.28059512912943673832341988762, −7.61246470342954427932063287488, −6.82003969865189276006949516718, −6.30784796667778428007013863282, −5.63801957412974851723597580673, −5.05456088714059487810990349873, −3.92742241754862252669855649525, −3.03767532819027704459668990231, −2.40736648500827413931436951360, −1.65874198879662468631758286874, −0.68342690084844621805365885683, 0.63032238224661034281072872003, 1.41858711176687961257747624312, 2.255367778512988978657255726913, 3.704317603710700857163567876610, 4.15256755274188632561049993099, 4.67187160006168751561569922993, 5.135775926842586698490207229634, 6.41862187063625135302726393635, 6.59607816298315661903555172884, 7.69041763523483563141692037665, 8.63125540588195388401618071182, 9.039051184039473333028569327105, 9.657007316240184369944408477631, 10.49432780067514482207400818679, 11.19077547557956756399270671418, 11.52436104776155723864928398371, 12.47702321806397363204647332588, 12.95083329516809953957011559047, 13.819898588750636701446047183031, 14.65570461336566545428831375138, 15.00206375054233804639422653694, 15.90524782628992450745680467319, 16.588749266963110039686125499741, 16.96791172563344136914343306954, 17.40494020450894020548667250139

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