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Properties

Label 1-4000-4000.1491-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.891 - 0.452i$
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  + (−0.790 − 0.612i)3-s + (−0.587 − 0.809i)7-s + (0.248 + 0.968i)9-s + (−0.995 − 0.0941i)11-s + (0.860 + 0.509i)13-s + (0.187 + 0.982i)17-s + (0.612 + 0.790i)19-s + (−0.0314 + 0.999i)21-s + (−0.844 − 0.535i)23-s + (0.397 − 0.917i)27-s + (−0.338 + 0.940i)29-s + (0.187 + 0.982i)31-s + (0.728 + 0.684i)33-s + (−0.917 + 0.397i)37-s + (−0.368 − 0.929i)39-s + ⋯
L(s)  = 1  + (−0.790 − 0.612i)3-s + (−0.587 − 0.809i)7-s + (0.248 + 0.968i)9-s + (−0.995 − 0.0941i)11-s + (0.860 + 0.509i)13-s + (0.187 + 0.982i)17-s + (0.612 + 0.790i)19-s + (−0.0314 + 0.999i)21-s + (−0.844 − 0.535i)23-s + (0.397 − 0.917i)27-s + (−0.338 + 0.940i)29-s + (0.187 + 0.982i)31-s + (0.728 + 0.684i)33-s + (−0.917 + 0.397i)37-s + (−0.368 − 0.929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.891 - 0.452i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.891 - 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03773166122 + 0.1577092473i\)
\(L(\frac12)\) \(\approx\) \(-0.03773166122 + 0.1577092473i\)
\(L(1)\) \(\approx\) \(0.6460333256 + 0.01620152342i\)
\(L(1)\) \(\approx\) \(0.6460333256 + 0.01620152342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.790 - 0.612i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.995 - 0.0941i)T \)
13 \( 1 + (0.860 + 0.509i)T \)
17 \( 1 + (0.187 + 0.982i)T \)
19 \( 1 + (0.612 + 0.790i)T \)
23 \( 1 + (-0.844 - 0.535i)T \)
29 \( 1 + (-0.338 + 0.940i)T \)
31 \( 1 + (0.187 + 0.982i)T \)
37 \( 1 + (-0.917 + 0.397i)T \)
41 \( 1 + (-0.844 + 0.535i)T \)
43 \( 1 + (-0.453 - 0.891i)T \)
47 \( 1 + (0.876 + 0.481i)T \)
53 \( 1 + (-0.0314 + 0.999i)T \)
59 \( 1 + (0.661 + 0.750i)T \)
61 \( 1 + (-0.975 + 0.218i)T \)
67 \( 1 + (-0.940 + 0.338i)T \)
71 \( 1 + (-0.481 + 0.876i)T \)
73 \( 1 + (-0.998 + 0.0627i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (0.612 + 0.790i)T \)
89 \( 1 + (0.998 - 0.0627i)T \)
97 \( 1 + (-0.425 - 0.904i)T \)
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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

−17.91823356028641884246956353826, −17.3567625222738016325164053631, −16.24510420213411060551848679758, −15.94119063800083210766127707196, −15.46100633305058515759229896100, −14.83203565095051055630971452445, −13.433604880022425938998946331639, −13.35788764539936978565244620077, −12.17577151722216695841563152577, −11.78833701902186486041820375997, −11.0544674221110618609293904490, −10.27863095255598781156444600067, −9.66825460395096681210520086298, −9.104752519613885217706642560783, −8.17882945859225772401301694807, −7.34417617909922814364312594161, −6.463802244517857762599701852602, −5.66298839383360028198825023159, −5.37380555075336834333072442588, −4.47976771834489792867317257386, −3.47168685762639022266555372763, −2.91895728210060322625823025784, −1.885547138664014520639098078260, −0.515084586539277712807604377864, −0.04849053324454988014886575493, 1.17868656507051910705605315612, 1.63385610819995920739052547873, 2.86909761712544165425434198740, 3.714431523413568156915214762116, 4.48631822477684782514973619253, 5.48454132245047384784024701091, 6.00465053926851764140991027469, 6.77400600257507297655326000067, 7.37505416870607808762757352058, 8.14528063569390959175174461497, 8.81718180658057705384656608022, 10.19726337706751941247871104460, 10.35704705363606213697266810349, 11.01082782339793766270239736473, 12.058803934676009353983978536482, 12.423833842937945647958940767280, 13.34021264007622797718343947596, 13.652085683865244532892683605886, 14.43212397318747139300416416131, 15.58098828151599833772950491109, 16.227980026358641502478370138453, 16.56723534275129706901467834072, 17.35543616340065919239051206356, 18.07392714265274553968248431853, 18.701433824577133949253320395315

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