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Properties

Label 1-4000-4000.1027-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  + (0.827 + 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

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

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: $\chi_{4000} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.0800 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.807312212 + 1.958222578i\)
\(L(\frac12)\) \(\approx\) \(1.807312212 + 1.958222578i\)
\(L(1)\) \(\approx\) \(1.448209478 + 0.5604365877i\)
\(L(1)\) \(\approx\) \(1.448209478 + 0.5604365877i\)

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

−18.309839725309947157390767870228, −17.900369448831837361669835624131, −16.83555806576186112740130182439, −16.39934577534148543402150596480, −15.47571684508293566709389581204, −14.65421560873438614819474935100, −14.1835630940030003797326732596, −13.61405565331535777871623565947, −13.0808462730969870153440621244, −11.986467170429333941421460139190, −11.69373041894410411587306311665, −10.45985938314592511158770720665, −10.244423756589793927957674682840, −8.92056254868793914058183365962, −8.54820926417683601339802014954, −7.89766046124469314002118100137, −7.21275534424966260522190193318, −6.46812294059425029535476724220, −5.6400478199061603053580641405, −4.70361448942113310618811008216, −3.722812656668236937242360181005, −3.36894832834320911548936564676, −2.127528637920539172866901841479, −1.60385675393187879816791480463, −0.66648164122709370455185432833, 1.22354668630486029002424969300, 2.03666425304898025201532658870, 2.91328649477965303928862200954, 3.406195295044102207034901343294, 4.53101297920529447099339864590, 5.21097450157107733577591047426, 5.5275490791177472646549311092, 7.09171496166083276168945196054, 7.525093643820668122256734418020, 8.32539791951454253482635687301, 8.83438297232505871422838825896, 9.82120224379602644967229779946, 10.066563998583746448380734293042, 11.15847460673553840816923196862, 11.630599271942649276872773116791, 12.64744680849701670196561673989, 13.25772654431064436568970854959, 14.0325829693001999029597676928, 14.63497377614750304892238266029, 15.330335233990633157977651385305, 15.71835820718749430631365448991, 16.38524095514815887341097892068, 17.495995503290210029939944284138, 18.072928371262505865844884704354, 18.49695680276191458192185415894

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