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Properties

Label 2-4000-100.19-c0-0-2
Degree 22
Conductor 40004000
Sign 0.944+0.327i0.944 + 0.327i
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  + (0.809 − 0.587i)3-s + 0.618·7-s + (0.587 + 0.190i)13-s + (−0.951 + 1.30i)19-s + (0.500 − 0.363i)21-s + (−0.309 − 0.951i)23-s + (0.309 + 0.951i)27-s + (1.30 − 0.951i)29-s + (0.587 − 0.809i)31-s + (0.951 + 0.309i)37-s + (0.587 − 0.190i)39-s + 1.61·43-s + (0.5 − 0.363i)47-s − 0.618·49-s + (0.587 + 0.809i)53-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + 0.618·7-s + (0.587 + 0.190i)13-s + (−0.951 + 1.30i)19-s + (0.500 − 0.363i)21-s + (−0.309 − 0.951i)23-s + (0.309 + 0.951i)27-s + (1.30 − 0.951i)29-s + (0.587 − 0.809i)31-s + (0.951 + 0.309i)37-s + (0.587 − 0.190i)39-s + 1.61·43-s + (0.5 − 0.363i)47-s − 0.618·49-s + (0.587 + 0.809i)53-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((0.944+0.327i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s) L(s)=((0.944+0.327i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 0.944+0.327i0.944 + 0.327i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4000(1599,⋅)\chi_{4000} (1599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :0), 0.944+0.327i)(2,\ 4000,\ (\ :0),\ 0.944 + 0.327i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8891626391.889162639
L(12)L(\frac12) ≈\approx 1.8891626391.889162639
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+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
7 1−0.618T+T2 1 - 0.618T + T^{2}
11 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
13 1+(−0.587−0.190i)T+(0.809+0.587i)T2 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2}
17 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
19 1+(0.951−1.30i)T+(−0.309−0.951i)T2 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2}
23 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
29 1+(−1.30+0.951i)T+(0.309−0.951i)T2 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}
31 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}
37 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
41 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
43 1−1.61T+T2 1 - 1.61T + T^{2}
47 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2}
53 1+(−0.587−0.809i)T+(−0.309+0.951i)T2 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2}
59 1+(1.53+0.5i)T+(0.809+0.587i)T2 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2}
61 1+(−0.309−0.951i)T+(−0.809+0.587i)T2 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}
67 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
71 1+(0.363+0.5i)T+(−0.309+0.951i)T2 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2}
73 1+(0.951−0.309i)T+(0.809−0.587i)T2 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2}
79 1+(0.951+1.30i)T+(−0.309+0.951i)T2 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2}
83 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
89 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
97 1+(0.363+0.5i)T+(−0.309+0.951i)T2 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)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

−8.430601477412977584209181480981, −7.969904573658160099611367251526, −7.39470215694039395949604184349, −6.27787217591316113712092730367, −5.91165909597146668772925994439, −4.56082731000933934018348751862, −4.12213048971615044049750113282, −2.86018878824578382324930078530, −2.20524270959923933514758482634, −1.25963153710636934195951980747, 1.20983687566452471654288351110, 2.51635316283310134377849970704, 3.15296087622758479518776470874, 4.16929355113858653926415632414, 4.64895125658658757164976916073, 5.65925443280982623871537556973, 6.49882962617836347972155090496, 7.28588477657767555184509712616, 8.257713957694964951337729259338, 8.589292359362662558456914078596

Graph of the ZZ-function along the critical line