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Properties

Label 1-4000-4000.3413-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.996 + 0.0800i$
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  + (0.278 + 0.960i)3-s + (−0.809 + 0.587i)7-s + (−0.844 + 0.535i)9-s + (−0.917 − 0.397i)11-s + (0.218 − 0.975i)13-s + (−0.684 + 0.728i)17-s + (0.278 − 0.960i)19-s + (−0.790 − 0.612i)21-s + (−0.637 − 0.770i)23-s + (−0.750 − 0.661i)27-s + (−0.562 − 0.827i)29-s + (0.728 + 0.684i)31-s + (0.125 − 0.992i)33-s + (0.661 + 0.750i)37-s + (0.998 − 0.0627i)39-s + ⋯
L(s)  = 1  + (0.278 + 0.960i)3-s + (−0.809 + 0.587i)7-s + (−0.844 + 0.535i)9-s + (−0.917 − 0.397i)11-s + (0.218 − 0.975i)13-s + (−0.684 + 0.728i)17-s + (0.278 − 0.960i)19-s + (−0.790 − 0.612i)21-s + (−0.637 − 0.770i)23-s + (−0.750 − 0.661i)27-s + (−0.562 − 0.827i)29-s + (0.728 + 0.684i)31-s + (0.125 − 0.992i)33-s + (0.661 + 0.750i)37-s + (0.998 − 0.0627i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.996 + 0.0800i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.996 + 0.0800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02619656664 + 0.6536600858i\)
\(L(\frac12)\) \(\approx\) \(0.02619656664 + 0.6536600858i\)
\(L(1)\) \(\approx\) \(0.8285436915 + 0.2795198270i\)
\(L(1)\) \(\approx\) \(0.8285436915 + 0.2795198270i\)

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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.917 - 0.397i)T \)
13 \( 1 + (0.218 - 0.975i)T \)
17 \( 1 + (-0.684 + 0.728i)T \)
19 \( 1 + (0.278 - 0.960i)T \)
23 \( 1 + (-0.637 - 0.770i)T \)
29 \( 1 + (-0.562 - 0.827i)T \)
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 \( 1 + (0.904 - 0.425i)T \)
53 \( 1 + (0.612 - 0.790i)T \)
59 \( 1 + (0.860 + 0.509i)T \)
61 \( 1 + (-0.995 - 0.0941i)T \)
67 \( 1 + (-0.562 + 0.827i)T \)
71 \( 1 + (0.904 - 0.425i)T \)
73 \( 1 + (0.968 + 0.248i)T \)
79 \( 1 + (0.876 + 0.481i)T \)
83 \( 1 + (-0.960 - 0.278i)T \)
89 \( 1 + (0.248 - 0.968i)T \)
97 \( 1 + (0.982 + 0.187i)T \)
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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

−18.22488623321666764815522037557, −17.36030670755793546951740091575, −16.66908190031399796488534216875, −15.95929793667732934049133507818, −15.3488868897793060177750068679, −14.24868590841512719900800600816, −13.82644851568333370558272795405, −13.278624139834425241438167718224, −12.54092220967809799340442946894, −12.00566636408860659699823412618, −11.13474688065682903950102616765, −10.4217057152976328228531524622, −9.40909062586681709970689035785, −9.12239575908175834078355081750, −7.9168768075308095461646508175, −7.47976391621304600565234080243, −6.85905551728959810824373745490, −6.08960724485242691831123927579, −5.4187772282388830749579739531, −4.21451150422700887379981823271, −3.60794119990751981636592246944, −2.58580807213791990518060809193, −2.0395771445492896331689565024, −1.00359486153245494693540717822, −0.13100553346669797944259345548, 0.71568431520099804769459556103, 2.444752850431019016692772106974, 2.68274426559281416857539281183, 3.54876134568256467647120668724, 4.375118729622335119979911354628, 5.17301474711466648868227945145, 5.88921190068599419746014660461, 6.44595759248266333843500106791, 7.70685390349114567036498632613, 8.35407778899063961938584348025, 8.8969629746966100321759457164, 9.74818744718392462107981667197, 10.28344107649781229969971089508, 10.9193239069444016028278362293, 11.62693347643590237551282535849, 12.64307678056752209648805113209, 13.229448992524235967117828638717, 13.76922025352026277760936423413, 14.901178910322044095443437778507, 15.288958153632139049800366031659, 15.89046421463684308839591799054, 16.327834307989300921205032019584, 17.239748685174770580586357464169, 17.968900452434864843290886523066, 18.63485723499303853792753759675

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