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Properties

Label 1-4000-4000.261-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.708 - 0.705i$
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  + (−0.338 − 0.940i)3-s + (0.587 − 0.809i)7-s + (−0.770 + 0.637i)9-s + (−0.750 + 0.661i)11-s + (−0.995 − 0.0941i)13-s + (0.992 − 0.125i)17-s + (−0.940 − 0.338i)19-s + (−0.960 − 0.278i)21-s + (−0.368 + 0.929i)23-s + (0.860 + 0.509i)27-s + (−0.999 + 0.0314i)29-s + (−0.992 + 0.125i)31-s + (0.876 + 0.481i)33-s + (0.509 + 0.860i)37-s + (0.248 + 0.968i)39-s + ⋯
L(s)  = 1  + (−0.338 − 0.940i)3-s + (0.587 − 0.809i)7-s + (−0.770 + 0.637i)9-s + (−0.750 + 0.661i)11-s + (−0.995 − 0.0941i)13-s + (0.992 − 0.125i)17-s + (−0.940 − 0.338i)19-s + (−0.960 − 0.278i)21-s + (−0.368 + 0.929i)23-s + (0.860 + 0.509i)27-s + (−0.999 + 0.0314i)29-s + (−0.992 + 0.125i)31-s + (0.876 + 0.481i)33-s + (0.509 + 0.860i)37-s + (0.248 + 0.968i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.708 - 0.705i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.708 - 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009881667 - 0.4173777413i\)
\(L(\frac12)\) \(\approx\) \(1.009881667 - 0.4173777413i\)
\(L(1)\) \(\approx\) \(0.8130687297 - 0.2544915717i\)
\(L(1)\) \(\approx\) \(0.8130687297 - 0.2544915717i\)

Euler product

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

−18.494521174205333462277303178212, −17.922677874679522406582699287515, −16.911510106130728642692503069929, −16.6770487310762072824477378033, −15.83303570382902415704659242632, −15.18521757203011751017342826194, −14.44501787513920082718295028197, −14.28677891695693429320521516888, −12.74976876834506499677856146513, −12.48566690534251795600515556640, −11.53066174847771921014568176136, −10.98570106743324807252849764294, −10.33247359241659687752445395870, −9.610095779350373930253733228118, −8.898220810489790156876951556170, −8.20911910208625850096997768210, −7.54615384617155358712176030225, −6.36540868465126588403158412176, −5.50920736649712848726203688676, −5.34208580595059518719627772779, −4.33141577097564566080833920372, −3.64454924954801952992532458039, −2.6238366411778079483448006880, −2.06009582778573337118842908242, −0.529018300236764076906583607408, 0.58306137859996858252481873134, 1.64403416406430342950707207605, 2.2047422395796981699287476230, 3.17360321418159766924476952090, 4.24718137156166433677093852308, 5.07206935637553009969599560247, 5.56361782247701468946711862753, 6.59378524702907826060411819841, 7.42646129337835079674654059575, 7.62346548632569976741514557719, 8.34383536278554468292378617336, 9.5294394574604956582161913357, 10.144827103037489741118212231485, 10.9921522149050924037686336537, 11.47431157294098244374408529825, 12.39450421560843515080162033541, 12.86715497643078338455854121477, 13.47650077939296916560521003922, 14.408465294251414795610438003941, 14.707287333602401906060365927268, 15.70135704513570201132457265732, 16.71358523047120049070921629727, 17.086447372756981889224502731687, 17.73288412408754689361134123206, 18.270109659698714740681490267836

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