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Properties

Label 2-4000-5.4-c1-0-28
Degree 22
Conductor 40004000
Sign −i-i
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.175i·3-s + 3.69i·7-s + 2.96·9-s + 0.648·21-s + 9.14i·23-s − 1.04i·27-s + 10.3·29-s − 12.7·41-s − 2.96i·43-s − 3.95i·47-s − 6.65·49-s + 6.15·61-s + 10.9i·63-s + 8.18i·67-s + 1.60·69-s + ⋯
L(s)  = 1  − 0.101i·3-s + 1.39i·7-s + 0.989·9-s + 0.141·21-s + 1.90i·23-s − 0.201i·27-s + 1.92·29-s − 1.99·41-s − 0.451i·43-s − 0.577i·47-s − 0.951·49-s + 0.787·61-s + 1.38i·63-s + 0.999i·67-s + 0.193·69-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: −i-i
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4000(1249,⋅)\chi_{4000} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), −i)(2,\ 4000,\ (\ :1/2),\ -i)

Particular Values

L(1)L(1) ≈\approx 1.8592832971.859283297
L(12)L(\frac12) ≈\approx 1.8592832971.859283297
L(32)L(\frac{3}{2}) 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+0.175iT−3T2 1 + 0.175iT - 3T^{2}
7 1−3.69iT−7T2 1 - 3.69iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−9.14iT−23T2 1 - 9.14iT - 23T^{2}
29 1−10.3T+29T2 1 - 10.3T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+12.7T+41T2 1 + 12.7T + 41T^{2}
43 1+2.96iT−43T2 1 + 2.96iT - 43T^{2}
47 1+3.95iT−47T2 1 + 3.95iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−6.15T+61T2 1 - 6.15T + 61T^{2}
67 1−8.18iT−67T2 1 - 8.18iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−16.8iT−83T2 1 - 16.8iT - 83T^{2}
89 1+5.66T+89T2 1 + 5.66T + 89T^{2}
97 1−97T2 1 - 97T^{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

−8.564299672327996903108788758562, −8.038194605984473392687238965889, −7.05126978821782211943391897693, −6.53402666873216640974815423955, −5.51901918797176949030565844084, −5.09082652579830570529980949913, −4.03432603270184418855965705381, −3.12711380461195135348024704427, −2.18069211587366132910647102342, −1.28644976561872794742822177659, 0.57372597763922028045587010865, 1.53061729892563565847636175894, 2.79760983656697653933147634410, 3.78450621781602904589294708336, 4.50626748564145703209401291594, 4.94233053711861396211075222202, 6.43169365744567370031207064480, 6.69563872788178179098668053295, 7.50209891694416844564853100011, 8.204108745552411177847737895346

Graph of the ZZ-function along the critical line