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Properties

Label 1-4000-4000.797-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.145 - 0.989i$
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  + (0.860 − 0.509i)3-s + (0.309 + 0.951i)7-s + (0.481 − 0.876i)9-s + (−0.827 − 0.562i)11-s + (−0.960 + 0.278i)13-s + (−0.368 + 0.929i)17-s + (0.860 + 0.509i)19-s + (0.750 + 0.661i)21-s + (−0.425 + 0.904i)23-s + (−0.0314 − 0.999i)27-s + (−0.0941 + 0.995i)29-s + (−0.929 − 0.368i)31-s + (−0.998 − 0.0627i)33-s + (−0.999 − 0.0314i)37-s + (−0.684 + 0.728i)39-s + ⋯
L(s)  = 1  + (0.860 − 0.509i)3-s + (0.309 + 0.951i)7-s + (0.481 − 0.876i)9-s + (−0.827 − 0.562i)11-s + (−0.960 + 0.278i)13-s + (−0.368 + 0.929i)17-s + (0.860 + 0.509i)19-s + (0.750 + 0.661i)21-s + (−0.425 + 0.904i)23-s + (−0.0314 − 0.999i)27-s + (−0.0941 + 0.995i)29-s + (−0.929 − 0.368i)31-s + (−0.998 − 0.0627i)33-s + (−0.999 − 0.0314i)37-s + (−0.684 + 0.728i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.145 - 0.989i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.145 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9893269727 - 1.145539884i\)
\(L(\frac12)\) \(\approx\) \(0.9893269727 - 1.145539884i\)
\(L(1)\) \(\approx\) \(1.195901305 - 0.09465170268i\)
\(L(1)\) \(\approx\) \(1.195901305 - 0.09465170268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.860 - 0.509i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.827 - 0.562i)T \)
13 \( 1 + (-0.960 + 0.278i)T \)
17 \( 1 + (-0.368 + 0.929i)T \)
19 \( 1 + (0.860 + 0.509i)T \)
23 \( 1 + (-0.425 + 0.904i)T \)
29 \( 1 + (-0.0941 + 0.995i)T \)
31 \( 1 + (-0.929 - 0.368i)T \)
37 \( 1 + (-0.999 - 0.0314i)T \)
41 \( 1 + (0.904 - 0.425i)T \)
43 \( 1 + (0.987 + 0.156i)T \)
47 \( 1 + (-0.844 - 0.535i)T \)
53 \( 1 + (0.661 - 0.750i)T \)
59 \( 1 + (-0.612 + 0.790i)T \)
61 \( 1 + (-0.940 + 0.338i)T \)
67 \( 1 + (-0.0941 - 0.995i)T \)
71 \( 1 + (-0.844 - 0.535i)T \)
73 \( 1 + (-0.992 - 0.125i)T \)
79 \( 1 + (0.968 + 0.248i)T \)
83 \( 1 + (-0.509 + 0.860i)T \)
89 \( 1 + (0.125 - 0.992i)T \)
97 \( 1 + (0.770 - 0.637i)T \)
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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

−18.48690071699109201350639118459, −17.79628634217115491353644884847, −17.191633344292675484130927945696, −16.19123904073046182639080073376, −15.87639991260814360155886658919, −15.02053551436020509835560227013, −14.38487085732837582251653440682, −13.85235785677675703264219662637, −13.17995678724409427250303841162, −12.49720667365640338450442773702, −11.51619441287695030855692603843, −10.68561081438510248890120095032, −10.178345791708877411273966665592, −9.53884683958694871133354906271, −8.87949885485313031552894917633, −7.72627919583018939867637784710, −7.60964329725973109076430378244, −6.868862082475843531141350502274, −5.53400707180348368011827436257, −4.67748352388038644771912363608, −4.45700430601889162975555510268, −3.33302532544692172709195814414, −2.62734235426766641299079454326, −1.97508699047358257788952264169, −0.74624771644983680141004714378, 0.217629214994864228175060611647, 1.6035483459265719579617147503, 2.01724809589485009102217180556, 2.951243172672601592340462282935, 3.52069303112887368668959662171, 4.57454219844550188667281092046, 5.54571843628589606380775428472, 5.98481743430836785675684958877, 7.19352216250289742155652712773, 7.61812249324084277847355083815, 8.39277168907261641375709464572, 8.987727546648078314741156304324, 9.617170032430702642396789433499, 10.47333898371268327987410736382, 11.351303323294759133908527289591, 12.189373275513606705808890879581, 12.587331226552889389330469501841, 13.38395289757672815649045030397, 14.08669642600253911877038157955, 14.699046094948522804105450377038, 15.29009948164979565855450373934, 15.93100494377391445424833860916, 16.720459590222823935152064811798, 17.8741284681288826208177180751, 18.052218192716088174302746681411

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