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Properties

Label 1-4000-4000.1109-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.0800 + 0.996i$
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  + (−0.562 − 0.827i)3-s + (0.587 + 0.809i)7-s + (−0.368 + 0.929i)9-s + (0.509 + 0.860i)11-s + (−0.917 + 0.397i)13-s + (0.876 − 0.481i)17-s + (0.827 + 0.562i)19-s + (0.338 − 0.940i)21-s + (−0.998 + 0.0627i)23-s + (0.975 − 0.218i)27-s + (−0.612 + 0.790i)29-s + (0.876 − 0.481i)31-s + (0.425 − 0.904i)33-s + (−0.218 + 0.975i)37-s + (0.844 + 0.535i)39-s + ⋯
L(s)  = 1  + (−0.562 − 0.827i)3-s + (0.587 + 0.809i)7-s + (−0.368 + 0.929i)9-s + (0.509 + 0.860i)11-s + (−0.917 + 0.397i)13-s + (0.876 − 0.481i)17-s + (0.827 + 0.562i)19-s + (0.338 − 0.940i)21-s + (−0.998 + 0.0627i)23-s + (0.975 − 0.218i)27-s + (−0.612 + 0.790i)29-s + (0.876 − 0.481i)31-s + (0.425 − 0.904i)33-s + (−0.218 + 0.975i)37-s + (0.844 + 0.535i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.0800 + 0.996i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.0800 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8715683446 + 0.8044009553i\)
\(L(\frac12)\) \(\approx\) \(0.8715683446 + 0.8044009553i\)
\(L(1)\) \(\approx\) \(0.9143354739 + 0.06573810432i\)
\(L(1)\) \(\approx\) \(0.9143354739 + 0.06573810432i\)

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 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (0.509 + 0.860i)T \)
13 \( 1 + (-0.917 + 0.397i)T \)
17 \( 1 + (0.876 - 0.481i)T \)
19 \( 1 + (0.827 + 0.562i)T \)
23 \( 1 + (-0.998 + 0.0627i)T \)
29 \( 1 + (-0.612 + 0.790i)T \)
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 \( 1 + (0.728 - 0.684i)T \)
53 \( 1 + (-0.338 + 0.940i)T \)
59 \( 1 + (0.995 - 0.0941i)T \)
61 \( 1 + (0.750 + 0.661i)T \)
67 \( 1 + (0.790 - 0.612i)T \)
71 \( 1 + (0.684 + 0.728i)T \)
73 \( 1 + (0.770 - 0.637i)T \)
79 \( 1 + (0.187 - 0.982i)T \)
83 \( 1 + (-0.827 - 0.562i)T \)
89 \( 1 + (0.770 - 0.637i)T \)
97 \( 1 + (0.992 - 0.125i)T \)
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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

−18.12303789227560680691339977276, −17.317967777278221101215816485504, −17.11752692403049105494063731460, −16.34011701202054797424846971446, −15.71618192462705971842752672457, −14.914748184916471883075200751264, −14.24327999176067212069973205477, −13.81564290241137563473041998081, −12.72904021855244036059399024646, −11.87428044060798638858014741441, −11.4758204863064094363084882970, −10.70563690818384216598524509911, −10.00525757327154134418392679142, −9.61892141367797292549411462905, −8.52136830455737909232450193541, −7.932477012331038426212836320348, −7.03595362768568498152674691859, −6.25992504598351147188817194673, −5.34406610547503441735147396559, −4.99331188924528914130991634164, −3.83042071607691197031487340496, −3.63567038128981556648913091265, −2.44864933810965191005957615969, −1.20004069485615929046997734062, −0.40839314922340119063970680282, 1.12856797265778696060165707816, 1.85208297948691850169155545765, 2.46266315570209536060523159035, 3.541110892702656248232370616825, 4.77031210104121165195621702478, 5.15122417134989749453183975813, 5.94958770873174846807136558769, 6.74142878654668016504175470963, 7.47532205627273610430670939376, 7.96645485988833231793537866898, 8.84408737741171481036128476792, 9.78956481445587572010959648902, 10.23094312403166107156182005231, 11.54087161920769259385403076214, 11.86881388363375823469757030458, 12.18689295042794647414389155013, 13.03483777292151071004404516674, 14.01732672571685003676661509308, 14.40420404671541344884768047418, 15.173845929736994346961572974772, 16.0178937379809763927997772202, 16.88691282511660104946861660147, 17.223772044621144773418112301095, 18.139286903298960041631175343067, 18.495031769835933919711250168444

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