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Properties

Label 2-4000-100.91-c0-0-0
Degree 22
Conductor 40004000
Sign −0.0314−0.999i-0.0314 - 0.999i
Analytic cond. 1.996261.99626
Root an. cond. 1.412891.41289
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)3-s + 1.61i·7-s + (−1.30 + 0.951i)13-s + (−0.587 − 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.587 + 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.190 − 0.587i)29-s + (0.951 + 0.309i)31-s + (−0.809 + 0.587i)37-s + (−0.951 + 1.30i)39-s − 0.618i·43-s + (1.53 − 0.5i)47-s − 1.61·49-s + (0.309 + 0.951i)53-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)3-s + 1.61i·7-s + (−1.30 + 0.951i)13-s + (−0.587 − 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.587 + 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.190 − 0.587i)29-s + (0.951 + 0.309i)31-s + (−0.809 + 0.587i)37-s + (−0.951 + 1.30i)39-s − 0.618i·43-s + (1.53 − 0.5i)47-s − 1.61·49-s + (0.309 + 0.951i)53-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((−0.0314−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0314 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s) L(s)=((−0.0314−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0314 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: −0.0314−0.999i-0.0314 - 0.999i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4000(1951,⋅)\chi_{4000} (1951, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :0), −0.0314−0.999i)(2,\ 4000,\ (\ :0),\ -0.0314 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3492303091.349230309
L(12)L(\frac12) ≈\approx 1.3492303091.349230309
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1+(−0.951+0.309i)T+(0.809−0.587i)T2 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2}
7 1−1.61iT−T2 1 - 1.61iT - T^{2}
11 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
13 1+(1.30−0.951i)T+(0.309−0.951i)T2 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2}
17 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
19 1+(0.587+0.190i)T+(0.809+0.587i)T2 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2}
23 1+(0.587−0.809i)T+(−0.309−0.951i)T2 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2}
29 1+(0.190+0.587i)T+(−0.809+0.587i)T2 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2}
31 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
37 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
41 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
43 1+0.618iT−T2 1 + 0.618iT - T^{2}
47 1+(−1.53+0.5i)T+(0.809−0.587i)T2 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2}
53 1+(−0.309−0.951i)T+(−0.809+0.587i)T2 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}
59 1+(−0.363−0.5i)T+(−0.309+0.951i)T2 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2}
61 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
67 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
71 1+(−1.53+0.5i)T+(0.809−0.587i)T2 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2}
73 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
79 1+(0.587−0.190i)T+(0.809−0.587i)T2 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2}
83 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
89 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
97 1+(0.5+1.53i)T+(−0.809+0.587i)T2 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.746491133000433485963573418523, −8.253595338696294372357407372546, −7.46124575027894671271054810777, −6.72845185826677989158284366430, −5.80147462905595997649701603856, −5.15902263361418326480020731209, −4.22706954287113224219582222954, −3.10860307334345410814658634908, −2.32371117744549810633837965577, −1.95519868848024717364140168301, 0.61141137689783561563809075641, 2.17167303544939574482900598708, 2.99886694531422314760539862061, 3.84792808343963239516333998619, 4.40461626678952350293642083608, 5.31796458978335664885111823196, 6.40662388691243263686420986219, 7.14483210249897194686944817386, 7.87253386156096891443956916207, 8.241029378551049914888964238678

Graph of the ZZ-function along the critical line