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Properties

Label 2-800-40.19-c4-0-55
Degree 22
Conductor 800800
Sign −0.673+0.739i-0.673 + 0.739i
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  − 7.09i·3-s − 2.59·7-s + 30.6·9-s + 7.95·11-s − 54.6·13-s + 28.7i·17-s − 316.·19-s + 18.3i·21-s + 846.·23-s − 792. i·27-s + 766. i·29-s − 1.19e3i·31-s − 56.4i·33-s − 2.43e3·37-s + 387. i·39-s + ⋯
L(s)  = 1  − 0.788i·3-s − 0.0528·7-s + 0.377·9-s + 0.0657·11-s − 0.323·13-s + 0.0995i·17-s − 0.877·19-s + 0.0417i·21-s + 1.60·23-s − 1.08i·27-s + 0.911i·29-s − 1.24i·31-s − 0.0518i·33-s − 1.77·37-s + 0.255i·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((−0.673+0.739i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+2) L(s)=((−0.673+0.739i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: −0.673+0.739i-0.673 + 0.739i
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), −0.673+0.739i)(2,\ 800,\ (\ :2),\ -0.673 + 0.739i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.6074607131.607460713
L(12)L(\frac12) ≈\approx 1.6074607131.607460713
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+7.09iT−81T2 1 + 7.09iT - 81T^{2}
7 1+2.59T+2.40e3T2 1 + 2.59T + 2.40e3T^{2}
11 1−7.95T+1.46e4T2 1 - 7.95T + 1.46e4T^{2}
13 1+54.6T+2.85e4T2 1 + 54.6T + 2.85e4T^{2}
17 1−28.7iT−8.35e4T2 1 - 28.7iT - 8.35e4T^{2}
19 1+316.T+1.30e5T2 1 + 316.T + 1.30e5T^{2}
23 1−846.T+2.79e5T2 1 - 846.T + 2.79e5T^{2}
29 1−766.iT−7.07e5T2 1 - 766. iT - 7.07e5T^{2}
31 1+1.19e3iT−9.23e5T2 1 + 1.19e3iT - 9.23e5T^{2}
37 1+2.43e3T+1.87e6T2 1 + 2.43e3T + 1.87e6T^{2}
41 1−2.43e3T+2.82e6T2 1 - 2.43e3T + 2.82e6T^{2}
43 1+2.33e3iT−3.41e6T2 1 + 2.33e3iT - 3.41e6T^{2}
47 1−2.19e3T+4.87e6T2 1 - 2.19e3T + 4.87e6T^{2}
53 1−3.04e3T+7.89e6T2 1 - 3.04e3T + 7.89e6T^{2}
59 1−455.T+1.21e7T2 1 - 455.T + 1.21e7T^{2}
61 1−3.92e3iT−1.38e7T2 1 - 3.92e3iT - 1.38e7T^{2}
67 1+4.82e3iT−2.01e7T2 1 + 4.82e3iT - 2.01e7T^{2}
71 1+874.iT−2.54e7T2 1 + 874. iT - 2.54e7T^{2}
73 1+7.68e3iT−2.83e7T2 1 + 7.68e3iT - 2.83e7T^{2}
79 1−1.47e3iT−3.89e7T2 1 - 1.47e3iT - 3.89e7T^{2}
83 1+6.84e3iT−4.74e7T2 1 + 6.84e3iT - 4.74e7T^{2}
89 1+1.31e4T+6.27e7T2 1 + 1.31e4T + 6.27e7T^{2}
97 1−8.60e3iT−8.85e7T2 1 - 8.60e3iT - 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

−9.248030542842477559148453613467, −8.533066802558389513877072537429, −7.41035476256194823091553213319, −6.98009859892635282294553590092, −5.99752428890600522838191680446, −4.91701170020515475683888163234, −3.86180600085002960354515015943, −2.56072042166678735985661021499, −1.54523078427723293385305670668, −0.39038261935295367369265931664, 1.16741015853439745720503248061, 2.59176451860281281571697753741, 3.72031245515733009994228526741, 4.60230225093739911855845234045, 5.36778439548858794857056743187, 6.61658368272782366164825622900, 7.33376490005029591051742409363, 8.518966080863109593382840070589, 9.224243557780943530330925319441, 10.03093108620269586416554418019

Graph of the ZZ-function along the critical line