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Properties

Label 1-4000-4000.1037-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.455 + 0.890i$
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  + (−0.562 − 0.827i)3-s + (−0.809 + 0.587i)7-s + (−0.368 + 0.929i)9-s + (−0.860 + 0.509i)11-s + (−0.917 + 0.397i)13-s + (−0.481 − 0.876i)17-s + (−0.562 + 0.827i)19-s + (0.940 + 0.338i)21-s + (0.0627 + 0.998i)23-s + (0.975 − 0.218i)27-s + (−0.790 − 0.612i)29-s + (0.876 − 0.481i)31-s + (0.904 + 0.425i)33-s + (−0.218 + 0.975i)37-s + (0.844 + 0.535i)39-s + ⋯
L(s)  = 1  + (−0.562 − 0.827i)3-s + (−0.809 + 0.587i)7-s + (−0.368 + 0.929i)9-s + (−0.860 + 0.509i)11-s + (−0.917 + 0.397i)13-s + (−0.481 − 0.876i)17-s + (−0.562 + 0.827i)19-s + (0.940 + 0.338i)21-s + (0.0627 + 0.998i)23-s + (0.975 − 0.218i)27-s + (−0.790 − 0.612i)29-s + (0.876 − 0.481i)31-s + (0.904 + 0.425i)33-s + (−0.218 + 0.975i)37-s + (0.844 + 0.535i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.455 + 0.890i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.455 + 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3155630112 + 0.1930365123i\)
\(L(\frac12)\) \(\approx\) \(0.3155630112 + 0.1930365123i\)
\(L(1)\) \(\approx\) \(0.5959289214 - 0.07949141828i\)
\(L(1)\) \(\approx\) \(0.5959289214 - 0.07949141828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.562 - 0.827i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.860 + 0.509i)T \)
13 \( 1 + (-0.917 + 0.397i)T \)
17 \( 1 + (-0.481 - 0.876i)T \)
19 \( 1 + (-0.562 + 0.827i)T \)
23 \( 1 + (0.0627 + 0.998i)T \)
29 \( 1 + (-0.790 - 0.612i)T \)
31 \( 1 + (0.876 - 0.481i)T \)
37 \( 1 + (-0.218 + 0.975i)T \)
41 \( 1 + (0.998 + 0.0627i)T \)
43 \( 1 + (0.891 - 0.453i)T \)
47 \( 1 + (-0.684 - 0.728i)T \)
53 \( 1 + (0.338 - 0.940i)T \)
59 \( 1 + (-0.0941 - 0.995i)T \)
61 \( 1 + (0.661 - 0.750i)T \)
67 \( 1 + (-0.790 + 0.612i)T \)
71 \( 1 + (-0.684 - 0.728i)T \)
73 \( 1 + (-0.637 - 0.770i)T \)
79 \( 1 + (-0.187 + 0.982i)T \)
83 \( 1 + (-0.827 - 0.562i)T \)
89 \( 1 + (0.770 - 0.637i)T \)
97 \( 1 + (-0.125 - 0.992i)T \)
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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

−17.89068569223378256599657654966, −17.4897886423230145513494485016, −16.744123517088713381737875781917, −16.17550838529497859435548262982, −15.63622247356589668747229852763, −14.88491582275110370133535926656, −14.286269320131983643379488714605, −13.17082645705523343375473146175, −12.792460880490402902556756094901, −12.07068114220401463866191988193, −10.86424035668489478178002888221, −10.73712974853758704421198319604, −10.07974150054283890583908808964, −9.23111436286541453494692828982, −8.6467348621298951052319650319, −7.61710029237036813280945449215, −6.852409971635743261859432404362, −6.08917659718011594996626046777, −5.4853772570731665355636357983, −4.50924923787679534706191829318, −4.11254823362680332016401142037, −2.99919083981387921796534934646, −2.5555686420745224764334525755, −0.95430415616880573685872228247, −0.14001282347134699169829519762, 0.42172916066916476712714320757, 1.81624920094339335540594178801, 2.33611019424681665972840871220, 3.08678264089428477125319491806, 4.32207587422794087498689915429, 5.12943185871743307887476632268, 5.752935132143476028657751580393, 6.4909723200145556053001109346, 7.20369069063518611784513303757, 7.7570204154009903984194481554, 8.59643488313823022090062814801, 9.631801262065008689174342176507, 9.98340085407623738039802551582, 11.00150442961714278896620543417, 11.8142487569769311966862225416, 12.15682941136830693874821835599, 13.09310745197649569403064453827, 13.280174439499658710529910550659, 14.27538991402121957626164021969, 15.11928522996554029045706386316, 15.804403945160427949616909986753, 16.4299983493956290766421103819, 17.20997293971714578799920903907, 17.731122709962893811991694653312, 18.52835240542867529103452171858

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