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Properties

Label 1-4000-4000.1181-r0-0-0
Degree $1$
Conductor $4000$
Sign $-0.997 - 0.0674i$
Analytic cond. $18.5759$
Root an. cond. $18.5759$
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  + (−0.218 + 0.975i)3-s + (−0.951 − 0.309i)7-s + (−0.904 − 0.425i)9-s + (0.0314 − 0.999i)11-s + (0.940 − 0.338i)13-s + (−0.0627 − 0.998i)17-s + (−0.975 + 0.218i)19-s + (0.509 − 0.860i)21-s + (−0.982 − 0.187i)23-s + (0.612 − 0.790i)27-s + (−0.397 − 0.917i)29-s + (0.0627 + 0.998i)31-s + (0.968 + 0.248i)33-s + (0.790 − 0.612i)37-s + (0.125 + 0.992i)39-s + ⋯
L(s)  = 1  + (−0.218 + 0.975i)3-s + (−0.951 − 0.309i)7-s + (−0.904 − 0.425i)9-s + (0.0314 − 0.999i)11-s + (0.940 − 0.338i)13-s + (−0.0627 − 0.998i)17-s + (−0.975 + 0.218i)19-s + (0.509 − 0.860i)21-s + (−0.982 − 0.187i)23-s + (0.612 − 0.790i)27-s + (−0.397 − 0.917i)29-s + (0.0627 + 0.998i)31-s + (0.968 + 0.248i)33-s + (0.790 − 0.612i)37-s + (0.125 + 0.992i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.997 - 0.0674i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.997 - 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001942096883 - 0.05748405522i\)
\(L(\frac12)\) \(\approx\) \(0.001942096883 - 0.05748405522i\)
\(L(1)\) \(\approx\) \(0.7243104844 + 0.06461158797i\)
\(L(1)\) \(\approx\) \(0.7243104844 + 0.06461158797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.218 + 0.975i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (0.0314 - 0.999i)T \)
13 \( 1 + (0.940 - 0.338i)T \)
17 \( 1 + (-0.0627 - 0.998i)T \)
19 \( 1 + (-0.975 + 0.218i)T \)
23 \( 1 + (-0.982 - 0.187i)T \)
29 \( 1 + (-0.397 - 0.917i)T \)
31 \( 1 + (0.0627 + 0.998i)T \)
37 \( 1 + (0.790 - 0.612i)T \)
41 \( 1 + (0.982 - 0.187i)T \)
43 \( 1 + (-0.987 + 0.156i)T \)
47 \( 1 + (0.637 - 0.770i)T \)
53 \( 1 + (0.509 - 0.860i)T \)
59 \( 1 + (-0.278 + 0.960i)T \)
61 \( 1 + (-0.827 + 0.562i)T \)
67 \( 1 + (-0.917 - 0.397i)T \)
71 \( 1 + (0.770 + 0.637i)T \)
73 \( 1 + (-0.481 + 0.876i)T \)
79 \( 1 + (-0.535 - 0.844i)T \)
83 \( 1 + (-0.975 + 0.218i)T \)
89 \( 1 + (0.481 - 0.876i)T \)
97 \( 1 + (-0.929 - 0.368i)T \)
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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

−18.67937120959023602495086447199, −18.30919555395056429139910352002, −17.489587121129420020481336196559, −16.85987725115259318584650999133, −16.24451950454209108882239726867, −15.32022944685443390162966389243, −14.817677595133528157867706986525, −13.82105304656789824926313028735, −13.23399420037367361022273343352, −12.598594955214594472656732348682, −12.255389922987353421586343975191, −11.28612003049917298791330121250, −10.67250879816332133068325163221, −9.77672614433375957051298123461, −9.03561062578200998910894745415, −8.30859329348544349284546675743, −7.588330942697319642548127060234, −6.75469238949438661349086625156, −6.17806221660796629046986491060, −5.78017561790831038474636303840, −4.52656648940531683874304563197, −3.80554057520103791642106249398, −2.802994049619250878846610186983, −2.00942185034889351838780757024, −1.363736780883114452290151688380, 0.019727613798359724902632496033, 0.90612048625453294540071035709, 2.42896593691591973056515982938, 3.182290337831322564770873097247, 3.853819161732631427782715452776, 4.40093568172423961687050835834, 5.59163795407990258553918863002, 5.99005947041882818300830166037, 6.670165549916056427174153354152, 7.75701314252765966352022755838, 8.66849494672773870311710122711, 9.06317869128177223394046127942, 10.036155323545929384145210966826, 10.41890636940921305436575965286, 11.19543213575897639132626237356, 11.77816072644749579899830394468, 12.70601850382163170358306355297, 13.51258836424561860036828262253, 13.98935139917289223928743405363, 14.86026711194837204001354076684, 15.65411178190823504833701081325, 16.21174492251187631543853036847, 16.495971135554393400375180592005, 17.3087808011650418981547341406, 18.180242864685261115333952559

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