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Properties

Label 1-4000-4000.1189-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.890 + 0.455i$
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  + (0.278 − 0.960i)3-s + (−0.587 + 0.809i)7-s + (−0.844 − 0.535i)9-s + (0.397 + 0.917i)11-s + (0.218 + 0.975i)13-s + (0.728 − 0.684i)17-s + (0.960 − 0.278i)19-s + (0.612 + 0.790i)21-s + (−0.770 − 0.637i)23-s + (−0.750 + 0.661i)27-s + (0.827 + 0.562i)29-s + (0.728 − 0.684i)31-s + (0.992 − 0.125i)33-s + (0.661 − 0.750i)37-s + (0.998 + 0.0627i)39-s + ⋯
L(s)  = 1  + (0.278 − 0.960i)3-s + (−0.587 + 0.809i)7-s + (−0.844 − 0.535i)9-s + (0.397 + 0.917i)11-s + (0.218 + 0.975i)13-s + (0.728 − 0.684i)17-s + (0.960 − 0.278i)19-s + (0.612 + 0.790i)21-s + (−0.770 − 0.637i)23-s + (−0.750 + 0.661i)27-s + (0.827 + 0.562i)29-s + (0.728 − 0.684i)31-s + (0.992 − 0.125i)33-s + (0.661 − 0.750i)37-s + (0.998 + 0.0627i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.890 + 0.455i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.890 + 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.604512995 + 0.3865425563i\)
\(L(\frac12)\) \(\approx\) \(1.604512995 + 0.3865425563i\)
\(L(1)\) \(\approx\) \(1.112504562 - 0.09483363495i\)
\(L(1)\) \(\approx\) \(1.112504562 - 0.09483363495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.278 - 0.960i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (0.397 + 0.917i)T \)
13 \( 1 + (0.218 + 0.975i)T \)
17 \( 1 + (0.728 - 0.684i)T \)
19 \( 1 + (0.960 - 0.278i)T \)
23 \( 1 + (-0.770 - 0.637i)T \)
29 \( 1 + (0.827 + 0.562i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (0.661 - 0.750i)T \)
41 \( 1 + (-0.770 + 0.637i)T \)
43 \( 1 + (-0.453 + 0.891i)T \)
47 \( 1 + (-0.425 + 0.904i)T \)
53 \( 1 + (-0.612 - 0.790i)T \)
59 \( 1 + (0.509 + 0.860i)T \)
61 \( 1 + (-0.0941 - 0.995i)T \)
67 \( 1 + (0.562 + 0.827i)T \)
71 \( 1 + (-0.904 - 0.425i)T \)
73 \( 1 + (0.248 + 0.968i)T \)
79 \( 1 + (-0.876 + 0.481i)T \)
83 \( 1 + (-0.960 + 0.278i)T \)
89 \( 1 + (0.248 + 0.968i)T \)
97 \( 1 + (0.187 + 0.982i)T \)
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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

−18.541972002318958792647680795135, −17.45270685149041383895715959971, −17.03681729760580340042052691279, −16.269958886713268084557586812453, −15.81315788810759788302229567727, −15.162701821969065158813929647242, −14.22167108055179566087317895132, −13.7926484665350458789735770863, −13.21872225835855200159529032617, −12.103127262597992338640675088063, −11.52829280616614075105600150661, −10.54613694928038446090611752468, −10.15690290378966788248565674092, −9.67264556205194136721759618809, −8.59848775123739135052991974592, −8.15424399822837189269081749909, −7.36060066165038789584826798227, −6.22846170251127082457252365970, −5.74369755973886867990146994159, −4.888107022494082151742573453624, −3.93654866716861172864827927628, −3.35320771179025301800795786771, −2.98669533306625733413533180848, −1.520722761367714430985381024117, −0.52495063319977350166799929241, 0.950122257827662538145231772181, 1.79295178299678014045136692868, 2.60066068928147541452901704992, 3.18293144379394070003057007384, 4.25849299198182369623940592435, 5.145004965229289297388911511656, 6.07351003867143963925216540672, 6.61835668267471173188073570571, 7.22831958617091626633374780117, 8.06940851446583111541404637275, 8.74274215206774747841001589580, 9.677760881965511065692088825651, 9.77253801784538245094256516454, 11.38792615586461466584789456683, 11.74916984016491927892950298917, 12.39963137064741159397959008083, 12.93930535129647244267075840747, 13.826038279597101654543380256662, 14.35715133659709200061040896619, 14.95099801816218050306966966708, 15.96856912960701114416614574109, 16.37751723180506007189165809491, 17.34098063161603233207384312557, 18.12880352714111637493559876591, 18.42673540521371979651726536387

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