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Properties

Label 1-4000-4000.613-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.650 - 0.759i$
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  + (−0.999 − 0.0314i)3-s + (−0.809 + 0.587i)7-s + (0.998 + 0.0627i)9-s + (−0.975 + 0.218i)11-s + (0.750 − 0.661i)13-s + (−0.904 − 0.425i)17-s + (−0.999 + 0.0314i)19-s + (0.827 − 0.562i)21-s + (0.968 + 0.248i)23-s + (−0.995 − 0.0941i)27-s + (0.960 − 0.278i)29-s + (−0.425 + 0.904i)31-s + (0.982 − 0.187i)33-s + (0.0941 + 0.995i)37-s + (−0.770 + 0.637i)39-s + ⋯
L(s)  = 1  + (−0.999 − 0.0314i)3-s + (−0.809 + 0.587i)7-s + (0.998 + 0.0627i)9-s + (−0.975 + 0.218i)11-s + (0.750 − 0.661i)13-s + (−0.904 − 0.425i)17-s + (−0.999 + 0.0314i)19-s + (0.827 − 0.562i)21-s + (0.968 + 0.248i)23-s + (−0.995 − 0.0941i)27-s + (0.960 − 0.278i)29-s + (−0.425 + 0.904i)31-s + (0.982 − 0.187i)33-s + (0.0941 + 0.995i)37-s + (−0.770 + 0.637i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.650 - 0.759i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.650 - 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07573517953 - 0.1646222468i\)
\(L(\frac12)\) \(\approx\) \(0.07573517953 - 0.1646222468i\)
\(L(1)\) \(\approx\) \(0.6131522736 + 0.03415078056i\)
\(L(1)\) \(\approx\) \(0.6131522736 + 0.03415078056i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.999 - 0.0314i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.975 + 0.218i)T \)
13 \( 1 + (0.750 - 0.661i)T \)
17 \( 1 + (-0.904 - 0.425i)T \)
19 \( 1 + (-0.999 + 0.0314i)T \)
23 \( 1 + (0.968 + 0.248i)T \)
29 \( 1 + (0.960 - 0.278i)T \)
31 \( 1 + (-0.425 + 0.904i)T \)
37 \( 1 + (0.0941 + 0.995i)T \)
41 \( 1 + (-0.248 - 0.968i)T \)
43 \( 1 + (-0.453 - 0.891i)T \)
47 \( 1 + (-0.125 - 0.992i)T \)
53 \( 1 + (0.562 + 0.827i)T \)
59 \( 1 + (0.397 + 0.917i)T \)
61 \( 1 + (-0.860 + 0.509i)T \)
67 \( 1 + (0.960 + 0.278i)T \)
71 \( 1 + (-0.125 - 0.992i)T \)
73 \( 1 + (-0.929 + 0.368i)T \)
79 \( 1 + (0.728 - 0.684i)T \)
83 \( 1 + (0.0314 + 0.999i)T \)
89 \( 1 + (0.368 + 0.929i)T \)
97 \( 1 + (0.481 - 0.876i)T \)
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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

−18.56768082456679174259651053302, −17.80120810477774779607381057825, −17.16254410279338959955804913626, −16.46945202519198754765803895389, −16.02704811127992109091806343489, −15.411471585890148915061616882803, −14.527401298345475084030305433580, −13.4550158617087467898479126992, −12.98324423760756785001360507793, −12.65516259859428124478001530736, −11.437262325431550165507871722797, −10.98308417741353219944685978331, −10.47342429325071272860841432067, −9.70792633473858868439933271858, −8.90389191784775238613833021259, −8.04334395978840243130054690585, −7.12627392234904930845310843124, −6.416304530542272802905176645406, −6.133363936579118396195152650968, −5.01944735182104437861100787843, −4.386492977367210501936599990513, −3.68084546435668193486613349758, −2.65819840033390405281997694587, −1.65079520888502005072715608851, −0.632019240344183707013092577271, 0.05488878399034087354186991390, 0.8919402156700140363679787694, 2.05965563879625826572576606957, 2.854509510965218414482502060799, 3.751275957378605000021197480583, 4.766431202855498698794560657076, 5.32369252679909896771782562506, 6.05084448085340577194769542849, 6.72441024894087842400713356094, 7.30074822653772745384633221265, 8.46048605122502374183856689591, 8.94049085241191938357662730954, 10.041050134052722642707496151524, 10.498545804219734463547982676305, 11.06810234332753809441029476005, 12.02949076931684052714195223334, 12.48350429796925941168924469713, 13.37537347353086497002425017989, 13.45349948448810909715934692941, 15.17097247832698401736138021086, 15.356684906657293416790579619379, 16.00836311660666719215328289790, 16.68201193067449584428404073383, 17.45203196595811812992261253180, 18.10254368089836439870577068035

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