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Properties

Label 2-4000-5.4-c1-0-16
Degree 22
Conductor 40004000
Sign 11
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  − 2.71i·3-s + 0.0156i·7-s − 4.35·9-s − 6.41·11-s + 2.76i·13-s − 2.07i·17-s + 4.25·19-s + 0.0424·21-s + 5.97i·23-s + 3.67i·27-s − 5.37·29-s + 3.59·31-s + 17.3i·33-s + 9.04i·37-s + 7.48·39-s + ⋯
L(s)  = 1  − 1.56i·3-s + 0.00591i·7-s − 1.45·9-s − 1.93·11-s + 0.765i·13-s − 0.503i·17-s + 0.976·19-s + 0.00926·21-s + 1.24i·23-s + 0.707i·27-s − 0.998·29-s + 0.645·31-s + 3.02i·33-s + 1.48i·37-s + 1.19·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 11
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), 1)(2,\ 4000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0869084541.086908454
L(12)L(\frac12) ≈\approx 1.0869084541.086908454
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+2.71iT−3T2 1 + 2.71iT - 3T^{2}
7 1−0.0156iT−7T2 1 - 0.0156iT - 7T^{2}
11 1+6.41T+11T2 1 + 6.41T + 11T^{2}
13 1−2.76iT−13T2 1 - 2.76iT - 13T^{2}
17 1+2.07iT−17T2 1 + 2.07iT - 17T^{2}
19 1−4.25T+19T2 1 - 4.25T + 19T^{2}
23 1−5.97iT−23T2 1 - 5.97iT - 23T^{2}
29 1+5.37T+29T2 1 + 5.37T + 29T^{2}
31 1−3.59T+31T2 1 - 3.59T + 31T^{2}
37 1−9.04iT−37T2 1 - 9.04iT - 37T^{2}
41 1+7.03T+41T2 1 + 7.03T + 41T^{2}
43 1+4.96iT−43T2 1 + 4.96iT - 43T^{2}
47 1+5.73iT−47T2 1 + 5.73iT - 47T^{2}
53 1−10.5iT−53T2 1 - 10.5iT - 53T^{2}
59 1−1.13T+59T2 1 - 1.13T + 59T^{2}
61 1−8.76T+61T2 1 - 8.76T + 61T^{2}
67 1+11.0iT−67T2 1 + 11.0iT - 67T^{2}
71 1−11.0T+71T2 1 - 11.0T + 71T^{2}
73 1−7.38iT−73T2 1 - 7.38iT - 73T^{2}
79 1−13.4T+79T2 1 - 13.4T + 79T^{2}
83 1−3.44iT−83T2 1 - 3.44iT - 83T^{2}
89 1−9.49T+89T2 1 - 9.49T + 89T^{2}
97 1−9.21iT−97T2 1 - 9.21iT - 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

−8.074732025343243265064283136266, −7.71001670641496577731026535644, −7.10438894906411783434809379114, −6.44723501377992077935440865674, −5.40896257680392037110426555917, −5.10214286674352807071033965776, −3.60424605129796427064705819963, −2.65733033454136379793378562369, −1.98504007878982449754107269802, −0.906175012772405247850480600087, 0.37408343939491008545430485415, 2.33630697861665353273623081850, 3.10161790044284023524734566302, 3.85428800822567848199330418508, 4.78455855587747458478360357638, 5.32742787850844154239825823883, 5.82946683123750682550523756503, 7.07994321190054923450041597110, 8.003875156362314393232657633812, 8.386706423706746547679962675994

Graph of the ZZ-function along the critical line