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Properties

Label 1-4000-4000.13-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.891 - 0.452i$
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  + (−0.612 + 0.790i)3-s + (0.809 − 0.587i)7-s + (−0.248 − 0.968i)9-s + (0.995 + 0.0941i)11-s + (−0.509 + 0.860i)13-s + (0.982 − 0.187i)17-s + (0.612 + 0.790i)19-s + (−0.0314 + 0.999i)21-s + (−0.535 + 0.844i)23-s + (0.917 + 0.397i)27-s + (0.338 − 0.940i)29-s + (−0.187 − 0.982i)31-s + (−0.684 + 0.728i)33-s + (0.397 + 0.917i)37-s + (−0.368 − 0.929i)39-s + ⋯
L(s)  = 1  + (−0.612 + 0.790i)3-s + (0.809 − 0.587i)7-s + (−0.248 − 0.968i)9-s + (0.995 + 0.0941i)11-s + (−0.509 + 0.860i)13-s + (0.982 − 0.187i)17-s + (0.612 + 0.790i)19-s + (−0.0314 + 0.999i)21-s + (−0.535 + 0.844i)23-s + (0.917 + 0.397i)27-s + (0.338 − 0.940i)29-s + (−0.187 − 0.982i)31-s + (−0.684 + 0.728i)33-s + (0.397 + 0.917i)37-s + (−0.368 − 0.929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.891 - 0.452i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.891 - 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07961011749 + 0.3327510981i\)
\(L(\frac12)\) \(\approx\) \(-0.07961011749 + 0.3327510981i\)
\(L(1)\) \(\approx\) \(0.8825538821 + 0.2300989527i\)
\(L(1)\) \(\approx\) \(0.8825538821 + 0.2300989527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.612 + 0.790i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.995 + 0.0941i)T \)
13 \( 1 + (-0.509 + 0.860i)T \)
17 \( 1 + (0.982 - 0.187i)T \)
19 \( 1 + (0.612 + 0.790i)T \)
23 \( 1 + (-0.535 + 0.844i)T \)
29 \( 1 + (0.338 - 0.940i)T \)
31 \( 1 + (-0.187 - 0.982i)T \)
37 \( 1 + (0.397 + 0.917i)T \)
41 \( 1 + (-0.844 + 0.535i)T \)
43 \( 1 + (-0.891 + 0.453i)T \)
47 \( 1 + (-0.481 + 0.876i)T \)
53 \( 1 + (-0.999 - 0.0314i)T \)
59 \( 1 + (0.661 + 0.750i)T \)
61 \( 1 + (-0.975 + 0.218i)T \)
67 \( 1 + (-0.338 - 0.940i)T \)
71 \( 1 + (0.481 - 0.876i)T \)
73 \( 1 + (-0.0627 - 0.998i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (0.790 - 0.612i)T \)
89 \( 1 + (-0.998 + 0.0627i)T \)
97 \( 1 + (-0.904 + 0.425i)T \)
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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

−17.92650107130060688447267436740, −17.38912915664230754889557497214, −16.71776894091143031188140381734, −16.02155379288286600696414927093, −15.11670783393862884375438669980, −14.318725562327917630178142671909, −14.03851591846622085318200520648, −12.88326097323426941327129578088, −12.34347521758326126601222503289, −11.86980426079353531076802673643, −11.182092883251706775199577989865, −10.47288277951540773542079065119, −9.658521087616950543672599763357, −8.55567521318895351219308034477, −8.25097775623067941750773172182, −7.233862544995693031816316749517, −6.79659259715101297835942352696, −5.76194188761888875076782134631, −5.29908275218929981051228657976, −4.633301680746227215333226078082, −3.42368244372915048208235510718, −2.58392139409214597909296603964, −1.65886547506949954391664225296, −1.046044823085599687808708401980, −0.05846226800830645228507741815, 1.1547231295699634201829674284, 1.67854275483365884021944067993, 3.11128369408825349363050884217, 3.86917961947820914976431817852, 4.47427368276727502930590625580, 5.09669042919426249285857345377, 6.00578134572357113849891827786, 6.60067832310330771631487151544, 7.61504868023403187922746764171, 8.125018684884308738228188708253, 9.37953759775029860780005395055, 9.66012362573002492353904185583, 10.3438284063132176838516646477, 11.29584192123467935464932492687, 11.89120862499370546873582489448, 11.990912498192785657755714702599, 13.413230495913080015677992317168, 14.117376707073591321744557597556, 14.655765813905733399269097154634, 15.20037563005927329439121374974, 16.222532855513278861495383005285, 16.79610217506694363773770690669, 17.07878065695153377737319041776, 17.89239699724214355457651927076, 18.56711010917811038066692101808

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