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Properties

Label 1-4000-4000.933-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.444 - 0.895i$
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  + (0.218 − 0.975i)3-s + (0.309 − 0.951i)7-s + (−0.904 − 0.425i)9-s + (−0.999 − 0.0314i)11-s + (−0.940 + 0.338i)13-s + (−0.998 + 0.0627i)17-s + (0.218 + 0.975i)19-s + (−0.860 − 0.509i)21-s + (−0.187 + 0.982i)23-s + (−0.612 + 0.790i)27-s + (−0.917 + 0.397i)29-s + (0.0627 + 0.998i)31-s + (−0.248 + 0.968i)33-s + (−0.790 + 0.612i)37-s + (0.125 + 0.992i)39-s + ⋯
L(s)  = 1  + (0.218 − 0.975i)3-s + (0.309 − 0.951i)7-s + (−0.904 − 0.425i)9-s + (−0.999 − 0.0314i)11-s + (−0.940 + 0.338i)13-s + (−0.998 + 0.0627i)17-s + (0.218 + 0.975i)19-s + (−0.860 − 0.509i)21-s + (−0.187 + 0.982i)23-s + (−0.612 + 0.790i)27-s + (−0.917 + 0.397i)29-s + (0.0627 + 0.998i)31-s + (−0.248 + 0.968i)33-s + (−0.790 + 0.612i)37-s + (0.125 + 0.992i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (933, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4664227826 - 0.2893634571i\)
\(L(\frac12)\) \(\approx\) \(0.4664227826 - 0.2893634571i\)
\(L(1)\) \(\approx\) \(0.7095036959 - 0.2636041121i\)
\(L(1)\) \(\approx\) \(0.7095036959 - 0.2636041121i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.218 - 0.975i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.999 - 0.0314i)T \)
13 \( 1 + (-0.940 + 0.338i)T \)
17 \( 1 + (-0.998 + 0.0627i)T \)
19 \( 1 + (0.218 + 0.975i)T \)
23 \( 1 + (-0.187 + 0.982i)T \)
29 \( 1 + (-0.917 + 0.397i)T \)
31 \( 1 + (0.0627 + 0.998i)T \)
37 \( 1 + (-0.790 + 0.612i)T \)
41 \( 1 + (-0.982 + 0.187i)T \)
43 \( 1 + (-0.987 + 0.156i)T \)
47 \( 1 + (-0.770 - 0.637i)T \)
53 \( 1 + (0.509 - 0.860i)T \)
59 \( 1 + (-0.960 - 0.278i)T \)
61 \( 1 + (-0.562 - 0.827i)T \)
67 \( 1 + (-0.917 - 0.397i)T \)
71 \( 1 + (-0.770 - 0.637i)T \)
73 \( 1 + (0.876 + 0.481i)T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (0.975 - 0.218i)T \)
89 \( 1 + (0.481 - 0.876i)T \)
97 \( 1 + (-0.368 + 0.929i)T \)
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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

−18.33382638363470901788028197196, −17.779731826578328169253409775956, −17.0082537452848536465852442622, −16.30677330074652045204732282320, −15.37292837650630657850542275546, −15.30908131960221086469127950969, −14.630974674953061071870683470131, −13.64148496842795806806129970444, −13.073919426496926883321972186457, −12.13385632521000062871814265880, −11.49730856028145473171438815706, −10.74399654385516765520406488984, −10.19282227853640080495419409667, −9.29935625594193817701163152935, −8.88064807602737835117996394246, −8.076157526182223744152773444099, −7.41043205089739418637427349996, −6.29244884678157969742495117529, −5.47163592884258739578988820863, −4.8996764216112918989285276256, −4.38153846614780734375197712153, −3.20686396761710806963417068160, −2.50751551547888852288901563929, −2.07525635876774227308456259962, −0.24620057617417386756767307952, 0.23957214120682238358495584606, 1.62829839395742181569240785569, 1.846162749947311071935967661285, 3.075134546413639229267387979147, 3.654572920276693036003644231342, 4.86962324587272425022796788906, 5.323795276256541502606215843987, 6.50189303420036745470460832972, 6.99026462414528205169533862282, 7.74368596473551097517840605917, 8.14203903410872085746162632823, 9.06910149397091954947888550194, 9.962010383155624266881119456, 10.607141441081421697837684780162, 11.43775901366659299352373389465, 12.04821429666810588022503802304, 12.83256362356651558681991762583, 13.51127796608392833652336674912, 13.863195452608769723912266360, 14.72053014523572036723575616461, 15.296784537774699894579650363997, 16.35250511960237611004102018087, 16.94335218690975789256547563169, 17.617915575650921071744558244408, 18.18710757342466956038941451966

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