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Properties

Label 1-4000-4000.1267-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.997 + 0.0674i$
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  + (0.0941 + 0.995i)3-s + (−0.309 + 0.951i)7-s + (−0.982 + 0.187i)9-s + (−0.612 + 0.790i)11-s + (−0.827 − 0.562i)13-s + (0.248 − 0.968i)17-s + (0.0941 − 0.995i)19-s + (−0.975 − 0.218i)21-s + (−0.728 + 0.684i)23-s + (−0.278 − 0.960i)27-s + (0.750 − 0.661i)29-s + (−0.968 − 0.248i)31-s + (−0.844 − 0.535i)33-s + (0.960 + 0.278i)37-s + (0.481 − 0.876i)39-s + ⋯
L(s)  = 1  + (0.0941 + 0.995i)3-s + (−0.309 + 0.951i)7-s + (−0.982 + 0.187i)9-s + (−0.612 + 0.790i)11-s + (−0.827 − 0.562i)13-s + (0.248 − 0.968i)17-s + (0.0941 − 0.995i)19-s + (−0.975 − 0.218i)21-s + (−0.728 + 0.684i)23-s + (−0.278 − 0.960i)27-s + (0.750 − 0.661i)29-s + (−0.968 − 0.248i)31-s + (−0.844 − 0.535i)33-s + (0.960 + 0.278i)37-s + (0.481 − 0.876i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.997 + 0.0674i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.997 + 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8923462534 + 0.03014788832i\)
\(L(\frac12)\) \(\approx\) \(0.8923462534 + 0.03014788832i\)
\(L(1)\) \(\approx\) \(0.7955763040 + 0.2944085855i\)
\(L(1)\) \(\approx\) \(0.7955763040 + 0.2944085855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.0941 + 0.995i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.612 + 0.790i)T \)
13 \( 1 + (-0.827 - 0.562i)T \)
17 \( 1 + (0.248 - 0.968i)T \)
19 \( 1 + (0.0941 - 0.995i)T \)
23 \( 1 + (-0.728 + 0.684i)T \)
29 \( 1 + (0.750 - 0.661i)T \)
31 \( 1 + (-0.968 - 0.248i)T \)
37 \( 1 + (0.960 + 0.278i)T \)
41 \( 1 + (-0.684 + 0.728i)T \)
43 \( 1 + (0.156 + 0.987i)T \)
47 \( 1 + (0.368 - 0.929i)T \)
53 \( 1 + (-0.218 + 0.975i)T \)
59 \( 1 + (0.338 + 0.940i)T \)
61 \( 1 + (-0.999 - 0.0314i)T \)
67 \( 1 + (-0.750 - 0.661i)T \)
71 \( 1 + (0.368 - 0.929i)T \)
73 \( 1 + (-0.425 - 0.904i)T \)
79 \( 1 + (0.637 - 0.770i)T \)
83 \( 1 + (0.995 + 0.0941i)T \)
89 \( 1 + (0.904 - 0.425i)T \)
97 \( 1 + (0.998 + 0.0627i)T \)
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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

−18.76451410371503408917609888509, −17.76815016323382098066183835960, −17.161251631320728698314599448308, −16.50060936190784187409552193528, −16.04272103817045099111938503047, −14.71932695407053118718905522021, −14.29041163549798925298250614797, −13.75426033970236174849151398486, −12.910718453579106237474079775808, −12.47560508283449188685413297235, −11.771338740875246713244496384665, −10.77627916282528580836740396832, −10.382881855887818985905336191036, −9.439874771557798249177917177626, −8.48902714487869265987445020033, −7.95133328692480591558065080991, −7.29758266865097153673570306690, −6.57842128308308730064898299510, −5.93635414623070685175669156622, −5.138787618305852566972169139492, −4.00834965220557430571255469035, −3.394406796437036393791609378362, −2.43775982229595174345442119471, −1.6630256203451379976948939534, −0.72835428808294608191874403230, 0.326425937202624068008787357062, 2.03004229998207341395555921509, 2.76139537986981711835598481974, 3.18931936699520164433957945398, 4.46825299055639479887550061852, 4.92206091861106319129620330780, 5.584182038786270992449495594619, 6.35180202098786613420945640663, 7.5203156297724154932735222862, 7.95899932510151113289990919699, 9.10248429598078483304584030672, 9.44882927202984964128025803158, 10.06716650913593475814764471004, 10.77419729489250404631800780486, 11.818315310768322956191253393261, 12.01887287708295648162961975181, 13.10937152752037804872632164495, 13.690985060914210982305120674888, 14.78133842421203648846637580228, 15.11237948829280180467153407155, 15.68927212641438119241284510085, 16.29413363601442982215198852973, 17.0438123442846249935805587956, 17.98960554875620968577057354026, 18.18554245831277775193326475223

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