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Properties

Label 2-800-40.19-c4-0-59
Degree 22
Conductor 800800
Sign −0.714+0.700i-0.714 + 0.700i
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  − 10.2i·3-s + 43.3·7-s − 23.1·9-s + 165.·11-s − 201.·13-s + 172. i·17-s − 640.·19-s − 441. i·21-s + 497.·23-s − 590. i·27-s − 509. i·29-s − 492. i·31-s − 1.68e3i·33-s + 678.·37-s + 2.05e3i·39-s + ⋯
L(s)  = 1  − 1.13i·3-s + 0.883·7-s − 0.285·9-s + 1.36·11-s − 1.19·13-s + 0.597i·17-s − 1.77·19-s − 1.00i·21-s + 0.940·23-s − 0.810i·27-s − 0.606i·29-s − 0.512i·31-s − 1.54i·33-s + 0.495·37-s + 1.35i·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((−0.714+0.700i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.700i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+2) L(s)=((−0.714+0.700i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: −0.714+0.700i-0.714 + 0.700i
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), −0.714+0.700i)(2,\ 800,\ (\ :2),\ -0.714 + 0.700i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.0473647642.047364764
L(12)L(\frac12) ≈\approx 2.0473647642.047364764
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+10.2iT−81T2 1 + 10.2iT - 81T^{2}
7 1−43.3T+2.40e3T2 1 - 43.3T + 2.40e3T^{2}
11 1−165.T+1.46e4T2 1 - 165.T + 1.46e4T^{2}
13 1+201.T+2.85e4T2 1 + 201.T + 2.85e4T^{2}
17 1−172.iT−8.35e4T2 1 - 172. iT - 8.35e4T^{2}
19 1+640.T+1.30e5T2 1 + 640.T + 1.30e5T^{2}
23 1−497.T+2.79e5T2 1 - 497.T + 2.79e5T^{2}
29 1+509.iT−7.07e5T2 1 + 509. iT - 7.07e5T^{2}
31 1+492.iT−9.23e5T2 1 + 492. iT - 9.23e5T^{2}
37 1−678.T+1.87e6T2 1 - 678.T + 1.87e6T^{2}
41 1+613.T+2.82e6T2 1 + 613.T + 2.82e6T^{2}
43 1+1.39e3iT−3.41e6T2 1 + 1.39e3iT - 3.41e6T^{2}
47 1−965.T+4.87e6T2 1 - 965.T + 4.87e6T^{2}
53 1−4.61e3T+7.89e6T2 1 - 4.61e3T + 7.89e6T^{2}
59 1+822.T+1.21e7T2 1 + 822.T + 1.21e7T^{2}
61 1+6.91e3iT−1.38e7T2 1 + 6.91e3iT - 1.38e7T^{2}
67 1+4.92e3iT−2.01e7T2 1 + 4.92e3iT - 2.01e7T^{2}
71 1+4.23e3iT−2.54e7T2 1 + 4.23e3iT - 2.54e7T^{2}
73 1−8.50e3iT−2.83e7T2 1 - 8.50e3iT - 2.83e7T^{2}
79 1+1.02e4iT−3.89e7T2 1 + 1.02e4iT - 3.89e7T^{2}
83 1+818.iT−4.74e7T2 1 + 818. iT - 4.74e7T^{2}
89 1−1.04e4T+6.27e7T2 1 - 1.04e4T + 6.27e7T^{2}
97 1+1.16e4iT−8.85e7T2 1 + 1.16e4iT - 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

−9.199615041360660596559620309586, −8.373735070085343036786289311816, −7.60241237695705864767860519251, −6.79095253603835812549211774312, −6.15156838463631012641659535679, −4.81832803346837291454019053832, −3.97779549028878599894719347715, −2.29717707242772159652673357047, −1.64444268524952570416178598654, −0.47854252105765561282699434614, 1.23432627196308448692049578344, 2.54962649969273788105077300777, 3.91024796753645282885444829957, 4.56852342436524208323713858479, 5.24626943964395500232285996501, 6.59814886274430914319353241886, 7.38412946074297020320509469152, 8.659032237291636822832427827387, 9.135216827976911894156515851170, 10.03518034208551245442515635389

Graph of the ZZ-function along the critical line