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Properties

Label 1-4000-4000.1187-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.705 - 0.708i$
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  + (0.940 + 0.338i)3-s + (0.809 − 0.587i)7-s + (0.770 + 0.637i)9-s + (−0.750 − 0.661i)11-s + (−0.0941 − 0.995i)13-s + (−0.125 + 0.992i)17-s + (0.940 − 0.338i)19-s + (0.960 − 0.278i)21-s + (0.929 − 0.368i)23-s + (0.509 + 0.860i)27-s + (−0.999 − 0.0314i)29-s + (0.992 + 0.125i)31-s + (−0.481 − 0.876i)33-s + (−0.860 − 0.509i)37-s + (0.248 − 0.968i)39-s + ⋯
L(s)  = 1  + (0.940 + 0.338i)3-s + (0.809 − 0.587i)7-s + (0.770 + 0.637i)9-s + (−0.750 − 0.661i)11-s + (−0.0941 − 0.995i)13-s + (−0.125 + 0.992i)17-s + (0.940 − 0.338i)19-s + (0.960 − 0.278i)21-s + (0.929 − 0.368i)23-s + (0.509 + 0.860i)27-s + (−0.999 − 0.0314i)29-s + (0.992 + 0.125i)31-s + (−0.481 − 0.876i)33-s + (−0.860 − 0.509i)37-s + (0.248 − 0.968i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.705 - 0.708i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.705 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.564567608 - 1.064639243i\)
\(L(\frac12)\) \(\approx\) \(2.564567608 - 1.064639243i\)
\(L(1)\) \(\approx\) \(1.592701166 - 0.1475420281i\)
\(L(1)\) \(\approx\) \(1.592701166 - 0.1475420281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.940 + 0.338i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-0.750 - 0.661i)T \)
13 \( 1 + (-0.0941 - 0.995i)T \)
17 \( 1 + (-0.125 + 0.992i)T \)
19 \( 1 + (0.940 - 0.338i)T \)
23 \( 1 + (0.929 - 0.368i)T \)
29 \( 1 + (-0.999 - 0.0314i)T \)
31 \( 1 + (0.992 + 0.125i)T \)
37 \( 1 + (-0.860 - 0.509i)T \)
41 \( 1 + (0.368 - 0.929i)T \)
43 \( 1 + (0.891 - 0.453i)T \)
47 \( 1 + (-0.982 - 0.187i)T \)
53 \( 1 + (-0.278 - 0.960i)T \)
59 \( 1 + (-0.975 - 0.218i)T \)
61 \( 1 + (-0.917 - 0.397i)T \)
67 \( 1 + (0.999 - 0.0314i)T \)
71 \( 1 + (-0.982 - 0.187i)T \)
73 \( 1 + (0.535 - 0.844i)T \)
79 \( 1 + (0.425 + 0.904i)T \)
83 \( 1 + (0.338 + 0.940i)T \)
89 \( 1 + (0.844 + 0.535i)T \)
97 \( 1 + (0.684 + 0.728i)T \)
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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

−18.77417161439574971254173201638, −17.99388046861392351101831332141, −17.46645741468722562623850142479, −16.39290732885980164606341211847, −15.65087346733121875784477156945, −15.135623800385035983331524449043, −14.43504507159734747886623018678, −13.86452908328306026910095532373, −13.23569966242322629311578836939, −12.40683653169172575571404822818, −11.77403590685258798815318789966, −11.15737720112911057705310993660, −10.06693103684084773006567121825, −9.33309784618841387341670046, −8.98611803952491990048303260532, −7.937718832654110808307254532164, −7.55370370161420241131710841299, −6.8734558537326967491435547823, −5.86464460246950072371647872969, −4.843152784284628735492850676606, −4.502561925781115577182512734858, −3.236084560646201916509122454701, −2.66536158570154731760496536780, −1.84995511711312667129684935520, −1.19783798978942976184495655290, 0.68964850419905848718949908807, 1.67592587225251780424306375488, 2.56128598830151095364159137185, 3.3459069283571365745573221301, 3.92400813809171625842930771102, 5.010527256358589668946991338019, 5.318746661842609395217512826077, 6.54518179293453711373778135890, 7.601813263586164968076058006433, 7.84251135273197609952589141399, 8.59726305877391374376877088607, 9.2656777975331262124657399664, 10.26869832494471175033312852516, 10.67869021172599790589617165264, 11.23922636816960314278176057742, 12.44945056778680875047581073186, 13.08793119986958657267410825433, 13.72225108690546747687427666675, 14.25614632416082439509103446856, 15.08690746636544725481721745473, 15.48277056276561413843153518869, 16.2418276479182370635892822568, 17.08759405219829489824032440796, 17.70709566357230092382027672947, 18.46095216685485897605524232004

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