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Properties

Label 1-4000-4000.1171-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.463 - 0.885i$
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  + (−0.661 + 0.750i)3-s + (−0.951 + 0.309i)7-s + (−0.125 − 0.992i)9-s + (0.338 + 0.940i)11-s + (−0.790 − 0.612i)13-s + (0.637 − 0.770i)17-s + (−0.750 + 0.661i)19-s + (0.397 − 0.917i)21-s + (−0.481 − 0.876i)23-s + (0.827 + 0.562i)27-s + (−0.975 + 0.218i)29-s + (0.637 − 0.770i)31-s + (−0.929 − 0.368i)33-s + (0.562 + 0.827i)37-s + (0.982 − 0.187i)39-s + ⋯
L(s)  = 1  + (−0.661 + 0.750i)3-s + (−0.951 + 0.309i)7-s + (−0.125 − 0.992i)9-s + (0.338 + 0.940i)11-s + (−0.790 − 0.612i)13-s + (0.637 − 0.770i)17-s + (−0.750 + 0.661i)19-s + (0.397 − 0.917i)21-s + (−0.481 − 0.876i)23-s + (0.827 + 0.562i)27-s + (−0.975 + 0.218i)29-s + (0.637 − 0.770i)31-s + (−0.929 − 0.368i)33-s + (0.562 + 0.827i)37-s + (0.982 − 0.187i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.463 - 0.885i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.463 - 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1753838399 - 0.1061534827i\)
\(L(\frac12)\) \(\approx\) \(0.1753838399 - 0.1061534827i\)
\(L(1)\) \(\approx\) \(0.6138792608 + 0.2005985857i\)
\(L(1)\) \(\approx\) \(0.6138792608 + 0.2005985857i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.661 + 0.750i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.338 + 0.940i)T \)
13 \( 1 + (-0.790 - 0.612i)T \)
17 \( 1 + (0.637 - 0.770i)T \)
19 \( 1 + (-0.750 + 0.661i)T \)
23 \( 1 + (-0.481 - 0.876i)T \)
29 \( 1 + (-0.975 + 0.218i)T \)
31 \( 1 + (0.637 - 0.770i)T \)
37 \( 1 + (0.562 + 0.827i)T \)
41 \( 1 + (-0.481 + 0.876i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + (0.968 + 0.248i)T \)
53 \( 1 + (0.397 - 0.917i)T \)
59 \( 1 + (-0.0314 + 0.999i)T \)
61 \( 1 + (-0.960 - 0.278i)T \)
67 \( 1 + (-0.218 + 0.975i)T \)
71 \( 1 + (-0.248 + 0.968i)T \)
73 \( 1 + (-0.684 + 0.728i)T \)
79 \( 1 + (0.0627 + 0.998i)T \)
83 \( 1 + (-0.750 + 0.661i)T \)
89 \( 1 + (0.684 - 0.728i)T \)
97 \( 1 + (0.535 - 0.844i)T \)
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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

−18.67344053628785709506057735499, −17.54361514960612508196034959027, −17.04774005895292590928229604276, −16.63196358906498728542872154081, −15.90130840047172058562206601864, −15.070397492507347121514609917296, −14.064786932752359762267827902743, −13.614219077521760913520947843784, −12.944358511265037259548890534519, −12.16775586689314995746187014804, −11.80040121660635351151785673336, −10.74650138416434700624609930888, −10.41343261387915816127457600265, −9.33825322395307600338497517465, −8.76271149611874115198017831214, −7.67083266476536793886900844443, −7.22829779321518751216461122116, −6.32296846741701104989919286793, −5.98009238724233637075544140784, −5.11130071920800030035230988281, −4.09375177583475607350065740166, −3.37340867679067236285876272253, −2.361219320229474258714786235731, −1.566931392664977319539912244300, −0.522069968888237406489986053665, 0.06162072273086810136357981252, 1.096729386503420963657769985825, 2.417072406829843886054517554305, 3.06600973777500740456564786867, 4.05664587838558416770623207503, 4.59895078587105443317250137477, 5.491847878712977833592126354524, 6.10826541088814887591053110294, 6.80459738123539487109286014730, 7.60506880915962444919735563336, 8.58333251171564364778793962246, 9.500078074875069631006242334059, 9.965396289931348613095409254263, 10.280157855615110308824369040644, 11.4351802050337617999647484325, 12.01500687575424441757153709153, 12.59689369463812980614999298206, 13.16068730764074582239115645285, 14.45287284594754085344880541485, 14.86913727843613110780228790407, 15.482187875857059603862672537664, 16.26417239755640363694572410863, 16.83669435732701513314110991066, 17.251810214424281858501280150000, 18.29301263395335287896566574948

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