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Properties

Label 1-4000-4000.1429-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.244 - 0.969i$
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  + (−0.860 + 0.509i)3-s + (0.951 − 0.309i)7-s + (0.481 − 0.876i)9-s + (0.562 − 0.827i)11-s + (0.960 − 0.278i)13-s + (−0.929 − 0.368i)17-s + (−0.509 + 0.860i)19-s + (−0.661 + 0.750i)21-s + (−0.904 − 0.425i)23-s + (0.0314 + 0.999i)27-s + (−0.995 − 0.0941i)29-s + (−0.929 − 0.368i)31-s + (−0.0627 + 0.998i)33-s + (0.999 + 0.0314i)37-s + (−0.684 + 0.728i)39-s + ⋯
L(s)  = 1  + (−0.860 + 0.509i)3-s + (0.951 − 0.309i)7-s + (0.481 − 0.876i)9-s + (0.562 − 0.827i)11-s + (0.960 − 0.278i)13-s + (−0.929 − 0.368i)17-s + (−0.509 + 0.860i)19-s + (−0.661 + 0.750i)21-s + (−0.904 − 0.425i)23-s + (0.0314 + 0.999i)27-s + (−0.995 − 0.0941i)29-s + (−0.929 − 0.368i)31-s + (−0.0627 + 0.998i)33-s + (0.999 + 0.0314i)37-s + (−0.684 + 0.728i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.244 - 0.969i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.244 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9360695973 - 0.7296290840i\)
\(L(\frac12)\) \(\approx\) \(0.9360695973 - 0.7296290840i\)
\(L(1)\) \(\approx\) \(0.8963158735 - 0.06841904069i\)
\(L(1)\) \(\approx\) \(0.8963158735 - 0.06841904069i\)

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 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.562 - 0.827i)T \)
13 \( 1 + (0.960 - 0.278i)T \)
17 \( 1 + (-0.929 - 0.368i)T \)
19 \( 1 + (-0.509 + 0.860i)T \)
23 \( 1 + (-0.904 - 0.425i)T \)
29 \( 1 + (-0.995 - 0.0941i)T \)
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 \( 1 + (0.535 - 0.844i)T \)
53 \( 1 + (0.661 - 0.750i)T \)
59 \( 1 + (0.790 + 0.612i)T \)
61 \( 1 + (0.338 + 0.940i)T \)
67 \( 1 + (-0.0941 - 0.995i)T \)
71 \( 1 + (0.844 + 0.535i)T \)
73 \( 1 + (0.125 - 0.992i)T \)
79 \( 1 + (-0.968 - 0.248i)T \)
83 \( 1 + (0.509 - 0.860i)T \)
89 \( 1 + (0.125 - 0.992i)T \)
97 \( 1 + (0.637 + 0.770i)T \)
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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

−18.43040395521733124202475320590, −17.869391139540515902259647670205, −17.419200406617247396618801094797, −16.813541345131442091988071695126, −15.85707793701972089346995655919, −15.3547744046561175341143053082, −14.52189947689301459748965668616, −13.78216434901950480817822298250, −13.03129506674000733646298842898, −12.448158461361622774977826749185, −11.64986512282313073341786713208, −11.113922000823424654481061390763, −10.72140677388551601683125444151, −9.58046682095004030523071999005, −8.85279875986834560346779079296, −8.11606456709661151977348608058, −7.28060278805733370005562270705, −6.70540526689649637418887303707, −5.91056891103142475020665168279, −5.282268963752552638119800040619, −4.351434386255107557682832023018, −3.95283867225436779708824515497, −2.29360236448905300097203489729, −1.86665924839362499880224974069, −1.040622114591465076368726730129, 0.42536483647449037468847116917, 1.344958147950884065717397308911, 2.223966084540342954763138500424, 3.73228248650998442433555662721, 3.92813454175794470809607930092, 4.82155407983408139665486474722, 5.74604223258114386267682174739, 6.104285532285176640361591810764, 7.00024197047440111038492084734, 7.9221859104044041715816654001, 8.66457728499274133552251458014, 9.27830515162466074305525233279, 10.36451155148541005996545358234, 10.74048726919159120150660329510, 11.559820607705603207761704765, 11.73202730482745560430872757464, 12.92631132325678114954429721272, 13.48598493921079026194969963069, 14.44078782862139975744161373945, 14.89662649272311397554455141734, 15.75794264164520041825283764032, 16.47163858096163063095008267411, 16.83768227327310475152755828172, 17.64758933366373186967556202106, 18.32545456953251531492451063212

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