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Properties

Label 1-4000-4000.1523-r0-0-0
Degree $1$
Conductor $4000$
Sign $-0.996 - 0.0800i$
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  + (−0.278 + 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

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

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.996 - 0.0800i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.996 - 0.0800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02970504370 + 0.7412040549i\)
\(L(\frac12)\) \(\approx\) \(0.02970504370 + 0.7412040549i\)
\(L(1)\) \(\approx\) \(0.6918033551 + 0.3526417046i\)
\(L(1)\) \(\approx\) \(0.6918033551 + 0.3526417046i\)

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

−18.15294097429129959981031264887, −17.79101374893507869651205074010, −16.62529881632472853542979484758, −16.24244646949203732142665081459, −15.51104868256639313368310768629, −14.70774056021880875193331333975, −13.83670154387021410090889453335, −13.09955227822505722734748430505, −12.871039314253199508141581627144, −11.99752013661976750297022858970, −11.44165789112894143064664232200, −10.52925859605745496289164778337, −9.95393027489105252289409398992, −8.89946675782968686631611905353, −8.33958461710959967680886021084, −7.555524022234640817649709189268, −6.93554385749163551542034923086, −6.06274509567285580829165176126, −5.53933770155690034714125394381, −4.93226434337696588261364167888, −3.5114160573645333752580125620, −2.7349339138632979646496501816, −2.38166905537019142191733961577, −0.96943610530061310542628769347, −0.27478938132159049156273779347, 1.09958107279733321831276850531, 2.2721347775058151035699502886, 3.22273074095741014657073364754, 3.971986945039370685417951236268, 4.35066793897446247620628332365, 5.60556289334516942004040789588, 5.85352539881786245383567124818, 6.87715673210243958051476833923, 7.66942226614042720899749593683, 8.443359456658190592996621117519, 9.43820006250270287276531767045, 9.862326231030276924204372079833, 10.44737647902569049902575217519, 11.09043237386669576916263986482, 11.993779601232378048599696786908, 12.56464388202986599663898994191, 13.43248427112078857471895949130, 14.19174836662877416543364967323, 14.72764308699173599103903062980, 15.69305632642583588117191109460, 16.16651497037952540724952400286, 16.5465594569805107845312353475, 17.40803930947929646528239621391, 17.976688930982732182027745796970, 18.98764344157663449269196509417

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