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Properties

Label 2-800-40.19-c4-0-35
Degree 22
Conductor 800800
Sign 0.0519−0.998i0.0519 - 0.998i
Analytic cond. 82.695982.6959
Root an. cond. 9.093739.09373
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.9i·3-s − 56.7·7-s − 173.·9-s + 122.·11-s + 77.6·13-s − 196. i·17-s + 400.·19-s − 904. i·21-s + 806.·23-s − 1.46e3i·27-s − 147. i·29-s − 1.05e3i·31-s + 1.95e3i·33-s + 1.21e3·37-s + 1.23e3i·39-s + ⋯
L(s)  = 1  + 1.77i·3-s − 1.15·7-s − 2.13·9-s + 1.01·11-s + 0.459·13-s − 0.679i·17-s + 1.10·19-s − 2.05i·21-s + 1.52·23-s − 2.01i·27-s − 0.175i·29-s − 1.09i·31-s + 1.79i·33-s + 0.885·37-s + 0.813i·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.0519−0.998i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0519 - 0.998i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+2) L(s)=((0.0519−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0519 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.0519−0.998i0.0519 - 0.998i
Analytic conductor: 82.695982.6959
Root analytic conductor: 9.093739.09373
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ800(399,⋅)\chi_{800} (399, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :2), 0.0519−0.998i)(2,\ 800,\ (\ :2),\ 0.0519 - 0.998i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.9657742371.965774237
L(12)L(\frac12) ≈\approx 1.9657742371.965774237
L(3)L(3) 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 1−15.9iT−81T2 1 - 15.9iT - 81T^{2}
7 1+56.7T+2.40e3T2 1 + 56.7T + 2.40e3T^{2}
11 1−122.T+1.46e4T2 1 - 122.T + 1.46e4T^{2}
13 1−77.6T+2.85e4T2 1 - 77.6T + 2.85e4T^{2}
17 1+196.iT−8.35e4T2 1 + 196. iT - 8.35e4T^{2}
19 1−400.T+1.30e5T2 1 - 400.T + 1.30e5T^{2}
23 1−806.T+2.79e5T2 1 - 806.T + 2.79e5T^{2}
29 1+147.iT−7.07e5T2 1 + 147. iT - 7.07e5T^{2}
31 1+1.05e3iT−9.23e5T2 1 + 1.05e3iT - 9.23e5T^{2}
37 1−1.21e3T+1.87e6T2 1 - 1.21e3T + 1.87e6T^{2}
41 1+2.41e3T+2.82e6T2 1 + 2.41e3T + 2.82e6T^{2}
43 1−922.iT−3.41e6T2 1 - 922. iT - 3.41e6T^{2}
47 1−3.72e3T+4.87e6T2 1 - 3.72e3T + 4.87e6T^{2}
53 1−3.43e3T+7.89e6T2 1 - 3.43e3T + 7.89e6T^{2}
59 1+1.90e3T+1.21e7T2 1 + 1.90e3T + 1.21e7T^{2}
61 1−700.iT−1.38e7T2 1 - 700. iT - 1.38e7T^{2}
67 1−1.05e3iT−2.01e7T2 1 - 1.05e3iT - 2.01e7T^{2}
71 1+6.07e3iT−2.54e7T2 1 + 6.07e3iT - 2.54e7T^{2}
73 1−2.12e3iT−2.83e7T2 1 - 2.12e3iT - 2.83e7T^{2}
79 1+5.44e3iT−3.89e7T2 1 + 5.44e3iT - 3.89e7T^{2}
83 1−9.58e3iT−4.74e7T2 1 - 9.58e3iT - 4.74e7T^{2}
89 1−4.84e3T+6.27e7T2 1 - 4.84e3T + 6.27e7T^{2}
97 1−2.61e3iT−8.85e7T2 1 - 2.61e3iT - 8.85e7T^{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

−9.600007818218926735489567862743, −9.435732817109693209595857949483, −8.650342838545689870819327010114, −7.22736158863225519239791601818, −6.20482404682867547078932217045, −5.35104744859431469046530081808, −4.35611005889812264655436047232, −3.52487854360384978000395147028, −2.86312447466292165706374029983, −0.72079954930435192913256239541, 0.74287211840348642794745126136, 1.46622860725953098769739139061, 2.80343957602274318478193069575, 3.63683018786208237002460249336, 5.39610590763646963095493769161, 6.33378526995271510400770432637, 6.82773254586713829431468028952, 7.50596110406468320074373210503, 8.661726742263366032088985897512, 9.166839975101157310206772355603

Graph of the ZZ-function along the critical line