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Properties

Label 1-4000-4000.3997-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.463 + 0.885i$
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  + (0.750 + 0.661i)3-s + (0.309 + 0.951i)7-s + (0.125 + 0.992i)9-s + (0.338 + 0.940i)11-s + (0.612 − 0.790i)13-s + (0.770 + 0.637i)17-s + (0.750 − 0.661i)19-s + (−0.397 + 0.917i)21-s + (0.876 − 0.481i)23-s + (−0.562 + 0.827i)27-s + (−0.975 + 0.218i)29-s + (−0.637 + 0.770i)31-s + (−0.368 + 0.929i)33-s + (0.827 − 0.562i)37-s + (0.982 − 0.187i)39-s + ⋯
L(s)  = 1  + (0.750 + 0.661i)3-s + (0.309 + 0.951i)7-s + (0.125 + 0.992i)9-s + (0.338 + 0.940i)11-s + (0.612 − 0.790i)13-s + (0.770 + 0.637i)17-s + (0.750 − 0.661i)19-s + (−0.397 + 0.917i)21-s + (0.876 − 0.481i)23-s + (−0.562 + 0.827i)27-s + (−0.975 + 0.218i)29-s + (−0.637 + 0.770i)31-s + (−0.368 + 0.929i)33-s + (0.827 − 0.562i)37-s + (0.982 − 0.187i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.463 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.463 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.463 + 0.885i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.463 + 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.210103673 + 3.651471989i\)
\(L(\frac12)\) \(\approx\) \(2.210103673 + 3.651471989i\)
\(L(1)\) \(\approx\) \(1.488943043 + 0.7540309519i\)
\(L(1)\) \(\approx\) \(1.488943043 + 0.7540309519i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.750 + 0.661i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.338 + 0.940i)T \)
13 \( 1 + (0.612 - 0.790i)T \)
17 \( 1 + (0.770 + 0.637i)T \)
19 \( 1 + (0.750 - 0.661i)T \)
23 \( 1 + (0.876 - 0.481i)T \)
29 \( 1 + (-0.975 + 0.218i)T \)
31 \( 1 + (-0.637 + 0.770i)T \)
37 \( 1 + (0.827 - 0.562i)T \)
41 \( 1 + (-0.481 + 0.876i)T \)
43 \( 1 + (0.987 + 0.156i)T \)
47 \( 1 + (0.248 - 0.968i)T \)
53 \( 1 + (0.917 + 0.397i)T \)
59 \( 1 + (0.0314 - 0.999i)T \)
61 \( 1 + (0.960 + 0.278i)T \)
67 \( 1 + (-0.975 - 0.218i)T \)
71 \( 1 + (0.248 - 0.968i)T \)
73 \( 1 + (0.728 + 0.684i)T \)
79 \( 1 + (0.0627 + 0.998i)T \)
83 \( 1 + (0.661 + 0.750i)T \)
89 \( 1 + (-0.684 + 0.728i)T \)
97 \( 1 + (0.844 + 0.535i)T \)
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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

−18.311334877173953265236670868861, −17.43071203406968042222399637942, −16.6559041776644185122726324453, −16.2641930044990632780531666723, −15.19974465411092517130185637577, −14.41591082936532250978416327862, −13.98867983110116125797389498045, −13.46040919440165767801321879529, −12.85555562232995142372357926429, −11.68668036552201201460968066122, −11.48337422642260456556966593654, −10.50735349425248139038835699436, −9.539494170125627053099443114445, −9.077762693675396348324909206278, −8.24096241248848573581268139144, −7.46926955342537372011444861371, −7.16449433729088753497974343545, −6.13634214577170398895178260162, −5.50863417671463296445703629070, −4.251302015650215852031366030067, −3.64295498476279722362901494235, −3.06720272987894603892825789250, −1.92674361532172023880720070243, −1.14499445578610224161181729525, −0.625544890776639717893098145, 1.019323255008147626229550750878, 1.93036983685961597096964435469, 2.70425831567087894222178133, 3.430187507678454920219661068897, 4.17537332297283888676425054889, 5.21065148967970505511893677661, 5.44753795927784361032187698522, 6.64015713714866966414853559957, 7.565822173538775441275981702931, 8.126829080869073007795600016878, 8.989127621486148730738084273409, 9.32562775579097507145281242339, 10.18445298911261826326548702358, 10.88058017594155464663344158262, 11.56582210228335289795026728859, 12.58827910914832910141769130645, 12.940353544006843591115818578918, 13.9190267702404133371293344990, 14.76379952304062444515973528230, 14.99926606273887302589703964405, 15.62839800438188037671676588088, 16.39730160795060149176715151414, 17.087539718931504174785564954893, 18.08582827474757215752094884811, 18.39755004515244447111156618197

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