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Properties

Label 1-4000-4000.589-r0-0-0
Degree $1$
Conductor $4000$
Sign $-0.759 - 0.650i$
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  + (0.0314 + 0.999i)3-s + (0.587 − 0.809i)7-s + (−0.998 + 0.0627i)9-s + (−0.975 − 0.218i)11-s + (0.661 − 0.750i)13-s + (−0.425 − 0.904i)17-s + (0.999 + 0.0314i)19-s + (0.827 + 0.562i)21-s + (−0.248 − 0.968i)23-s + (−0.0941 − 0.995i)27-s + (−0.960 − 0.278i)29-s + (−0.425 − 0.904i)31-s + (0.187 − 0.982i)33-s + (0.995 + 0.0941i)37-s + (0.770 + 0.637i)39-s + ⋯
L(s)  = 1  + (0.0314 + 0.999i)3-s + (0.587 − 0.809i)7-s + (−0.998 + 0.0627i)9-s + (−0.975 − 0.218i)11-s + (0.661 − 0.750i)13-s + (−0.425 − 0.904i)17-s + (0.999 + 0.0314i)19-s + (0.827 + 0.562i)21-s + (−0.248 − 0.968i)23-s + (−0.0941 − 0.995i)27-s + (−0.960 − 0.278i)29-s + (−0.425 − 0.904i)31-s + (0.187 − 0.982i)33-s + (0.995 + 0.0941i)37-s + (0.770 + 0.637i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.759 - 0.650i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1701517399 - 0.4601033148i\)
\(L(\frac12)\) \(\approx\) \(0.1701517399 - 0.4601033148i\)
\(L(1)\) \(\approx\) \(0.8952605010 + 0.03932431678i\)
\(L(1)\) \(\approx\) \(0.8952605010 + 0.03932431678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.0314 + 0.999i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (-0.975 - 0.218i)T \)
13 \( 1 + (0.661 - 0.750i)T \)
17 \( 1 + (-0.425 - 0.904i)T \)
19 \( 1 + (0.999 + 0.0314i)T \)
23 \( 1 + (-0.248 - 0.968i)T \)
29 \( 1 + (-0.960 - 0.278i)T \)
31 \( 1 + (-0.425 - 0.904i)T \)
37 \( 1 + (0.995 + 0.0941i)T \)
41 \( 1 + (-0.248 + 0.968i)T \)
43 \( 1 + (0.891 + 0.453i)T \)
47 \( 1 + (-0.992 - 0.125i)T \)
53 \( 1 + (-0.827 - 0.562i)T \)
59 \( 1 + (-0.397 + 0.917i)T \)
61 \( 1 + (-0.860 - 0.509i)T \)
67 \( 1 + (0.278 + 0.960i)T \)
71 \( 1 + (-0.125 + 0.992i)T \)
73 \( 1 + (-0.368 + 0.929i)T \)
79 \( 1 + (-0.728 - 0.684i)T \)
83 \( 1 + (-0.999 - 0.0314i)T \)
89 \( 1 + (-0.368 + 0.929i)T \)
97 \( 1 + (-0.876 + 0.481i)T \)
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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

−18.594302998987871685295484842990, −18.1398817681590827042265493746, −17.667215993630339655145654265733, −16.84312773887474679554786180035, −15.91846182258973827684668588242, −15.38083184046992961996231950933, −14.54751139398495989003188073438, −13.90135114044480294164552681456, −13.26679707963204002595871399766, −12.56028228306636179904638685071, −12.0116231861110762602169234103, −11.15589819981049549234072528493, −10.84582004544213060861052005489, −9.49760996289130386439979155574, −8.95496933278549992699037635115, −8.15542954580574920895421705161, −7.64411017224308423751258866110, −6.911598469042984300316107119083, −5.92070330125576905444682280347, −5.565097848862242967058806148412, −4.66692913294515116921807031981, −3.53759784801027495978973686041, −2.752285914641724772547123106895, −1.76113708814795354654825231645, −1.520346355769426766291989252986, 0.134055008412909456991192946527, 1.17057011945459015212241255540, 2.53229673550834443141774878842, 3.090008065779969516476180244020, 4.039039461036503376266642245, 4.61750941308202547853318506440, 5.39134347294555769042050382871, 5.95932640244135542128123099048, 7.11475216177384213339921436037, 7.98387669322720714569147764721, 8.27551624487512838009895861768, 9.44250747977846647887966971924, 9.86553196520354227392825301284, 10.76618992014573742455186888047, 11.114710164331099272833538247122, 11.72410284392838449207753622745, 13.03661153925280378883389632415, 13.40990162921257845197717707666, 14.26980738008397792335909615478, 14.81355279821883941430189348281, 15.609654430173786309618393205032, 16.20531591716154792378736010866, 16.63247951005181393260004650100, 17.57759876251354213255507222879, 18.13112844761338817198028897231

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