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Properties

Label 1-4000-4000.723-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  + (0.827 + 0.562i)3-s + (−0.809 − 0.587i)7-s + (0.368 + 0.929i)9-s + (0.509 − 0.860i)11-s + (0.397 − 0.917i)13-s + (−0.481 + 0.876i)17-s + (−0.827 + 0.562i)19-s + (−0.338 − 0.940i)21-s + (0.0627 − 0.998i)23-s + (−0.218 + 0.975i)27-s + (−0.612 − 0.790i)29-s + (−0.876 − 0.481i)31-s + (0.904 − 0.425i)33-s + (−0.975 + 0.218i)37-s + (0.844 − 0.535i)39-s + ⋯
L(s)  = 1  + (0.827 + 0.562i)3-s + (−0.809 − 0.587i)7-s + (0.368 + 0.929i)9-s + (0.509 − 0.860i)11-s + (0.397 − 0.917i)13-s + (−0.481 + 0.876i)17-s + (−0.827 + 0.562i)19-s + (−0.338 − 0.940i)21-s + (0.0627 − 0.998i)23-s + (−0.218 + 0.975i)27-s + (−0.612 − 0.790i)29-s + (−0.876 − 0.481i)31-s + (0.904 − 0.425i)33-s + (−0.975 + 0.218i)37-s + (0.844 − 0.535i)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: $\chi_{4000} (723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.996 - 0.0800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001480452732 + 0.03694044618i\)
\(L(\frac12)\) \(\approx\) \(0.001480452732 + 0.03694044618i\)
\(L(1)\) \(\approx\) \(1.024486841 + 0.07649693417i\)
\(L(1)\) \(\approx\) \(1.024486841 + 0.07649693417i\)

Euler product

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

−18.32346600910388801152411306688, −17.59843635462816010041627233514, −16.78905937956978217760581814028, −15.9494287980773787176973293749, −15.270483460667172348974956620390, −14.86172030751930374948955759761, −13.75094743646627831924411331963, −13.60393350299767811114794749095, −12.54971812120122616289625507219, −12.22841835652264859099836268703, −11.41653019637649726309897717065, −10.459760667354852287435754459603, −9.3714833032213775407572025930, −9.17354128849772277050619001188, −8.641463282160869133820038512829, −7.405977353595664799432214314, −6.95559330710186799138304974421, −6.44673611112354663211029114970, −5.43151640789870308618736740106, −4.440960095430834254983406356699, −3.63375268195324570920926040650, −2.976849593731283615102544146345, −1.98235095643487790947106601676, −1.60331028169226525039365601505, −0.00829694762680476724440609685, 1.304182356624237684854811587391, 2.30588301658417611543015167593, 3.18216231529768699567769502694, 3.82242125911828222868760643092, 4.22147740498888001772320787747, 5.4410999810657903987951088135, 6.18241511450261916864146366591, 6.882346124801444913181876354448, 7.89918260439152683402970098011, 8.45772423484655311482449774942, 9.028527652262341177439147789255, 9.89581154116798039278187251856, 10.59217710234386813790685912011, 10.84788827201698070604100810209, 12.06027444709534612139462937757, 13.02306576498954205743818672226, 13.31188876384472989422284771069, 14.03694503042191615493966627228, 14.93587847534771410740825377018, 15.23483101821770484482823090100, 16.1816456202805837884168943362, 16.72267365119764015448516385606, 17.16607281342559777271847675472, 18.40501531275391432104396811427, 18.95438216334832360949625764816

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