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Properties

Label 1-4000-4000.979-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.145 - 0.989i$
Analytic cond. $429.859$
Root an. cond. $429.859$
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  + (0.917 + 0.397i)3-s + (0.951 − 0.309i)7-s + (0.684 + 0.728i)9-s + (0.278 − 0.960i)11-s + (−0.999 − 0.0314i)13-s + (0.535 + 0.844i)17-s + (0.397 + 0.917i)19-s + (0.995 + 0.0941i)21-s + (0.125 − 0.992i)23-s + (0.338 + 0.940i)27-s + (−0.509 − 0.860i)29-s + (−0.535 − 0.844i)31-s + (0.637 − 0.770i)33-s + (0.940 + 0.338i)37-s + (−0.904 − 0.425i)39-s + ⋯
L(s)  = 1  + (0.917 + 0.397i)3-s + (0.951 − 0.309i)7-s + (0.684 + 0.728i)9-s + (0.278 − 0.960i)11-s + (−0.999 − 0.0314i)13-s + (0.535 + 0.844i)17-s + (0.397 + 0.917i)19-s + (0.995 + 0.0941i)21-s + (0.125 − 0.992i)23-s + (0.338 + 0.940i)27-s + (−0.509 − 0.860i)29-s + (−0.535 − 0.844i)31-s + (0.637 − 0.770i)33-s + (0.940 + 0.338i)37-s + (−0.904 − 0.425i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.145 - 0.989i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.145 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.741527883 - 2.016511937i\)
\(L(\frac12)\) \(\approx\) \(1.741527883 - 2.016511937i\)
\(L(1)\) \(\approx\) \(1.513877389 - 0.07130611875i\)
\(L(1)\) \(\approx\) \(1.513877389 - 0.07130611875i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.917 + 0.397i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (-0.999 - 0.0314i)T \)
17 \( 1 + (0.535 + 0.844i)T \)
19 \( 1 + (0.397 + 0.917i)T \)
23 \( 1 + (0.125 - 0.992i)T \)
29 \( 1 + (-0.509 - 0.860i)T \)
31 \( 1 + (-0.535 - 0.844i)T \)
37 \( 1 + (0.940 + 0.338i)T \)
41 \( 1 + (-0.125 - 0.992i)T \)
43 \( 1 + (0.156 - 0.987i)T \)
47 \( 1 + (-0.0627 - 0.998i)T \)
53 \( 1 + (-0.995 - 0.0941i)T \)
59 \( 1 + (-0.562 - 0.827i)T \)
61 \( 1 + (0.612 + 0.790i)T \)
67 \( 1 + (-0.860 - 0.509i)T \)
71 \( 1 + (-0.998 + 0.0627i)T \)
73 \( 1 + (-0.982 + 0.187i)T \)
79 \( 1 + (-0.929 + 0.368i)T \)
83 \( 1 + (-0.397 - 0.917i)T \)
89 \( 1 + (-0.982 + 0.187i)T \)
97 \( 1 + (-0.968 - 0.248i)T \)
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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

−18.40577897624960719426442486914, −17.84306360882255157119643808617, −17.48013760401010345346139058138, −16.38386473811053173325327602365, −15.606169405801344682156555159375, −14.777000503223866608153216255012, −14.59486567242882435005007130241, −13.88206049680447395167574102851, −12.98836770612680499362921912207, −12.421481270003430761510087204930, −11.70946323846518276837338380651, −11.085890141737067075603806933627, −9.8765275845376676700286456610, −9.428909316837633999054879498781, −8.86921307266367870958478240696, −7.81632960740844253480643727577, −7.422513447868387638210553470185, −6.92352749473351295542112145342, −5.70540971321988799646941519132, −4.81118549870332008278381935738, −4.40208766526128855695073877594, −3.11270138084915287392850462445, −2.67020746706390771684152871180, −1.64987449367191127862454246896, −1.19614215951038295882072787460, 0.29721900959337849570396717098, 1.43547438322982290453887711910, 2.1227543830199141564935226296, 3.005109275489694191757312230546, 3.88618246233352486674058869174, 4.34170247720299288205536774428, 5.32095022783157541537704160658, 5.98006519828222572913258902880, 7.194790417017454161451521414141, 7.782041063656221391985935062766, 8.338647890142281364472980138784, 8.96933877240812740746321034534, 9.911693309272963174706300211630, 10.36828094756131568878161390589, 11.1661128550534195711341501654, 11.91627302796936737030091826242, 12.74169515950760261096517389030, 13.54533156975913319723840544939, 14.212038021374271209786666212906, 14.685073385610217580239188162921, 15.12017529810336224492468159346, 16.13008999549398059849438874681, 16.86615475727963160494852985958, 17.15192917888521481282012095998, 18.376525804800139878751254109654

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