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Properties

Label 2-800-40.19-c4-0-9
Degree 22
Conductor 800800
Sign 0.999+0.0116i0.999 + 0.0116i
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  − 13.1i·3-s − 61.6·7-s − 91.5·9-s − 182.·11-s + 141.·13-s − 40.3i·17-s − 321.·19-s + 809. i·21-s − 816.·23-s + 138. i·27-s − 1.19e3i·29-s + 361. i·31-s + 2.39e3i·33-s − 0.0209·37-s − 1.86e3i·39-s + ⋯
L(s)  = 1  − 1.45i·3-s − 1.25·7-s − 1.12·9-s − 1.50·11-s + 0.838·13-s − 0.139i·17-s − 0.890·19-s + 1.83i·21-s − 1.54·23-s + 0.189i·27-s − 1.42i·29-s + 0.376i·31-s + 2.19i·33-s − 1.52e − 5·37-s − 1.22i·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.999+0.0116i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0116i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+2) L(s)=((0.999+0.0116i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 + 0.0116i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.999+0.0116i0.999 + 0.0116i
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), 0.999+0.0116i)(2,\ 800,\ (\ :2),\ 0.999 + 0.0116i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.59682612150.5968261215
L(12)L(\frac12) ≈\approx 0.59682612150.5968261215
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+13.1iT−81T2 1 + 13.1iT - 81T^{2}
7 1+61.6T+2.40e3T2 1 + 61.6T + 2.40e3T^{2}
11 1+182.T+1.46e4T2 1 + 182.T + 1.46e4T^{2}
13 1−141.T+2.85e4T2 1 - 141.T + 2.85e4T^{2}
17 1+40.3iT−8.35e4T2 1 + 40.3iT - 8.35e4T^{2}
19 1+321.T+1.30e5T2 1 + 321.T + 1.30e5T^{2}
23 1+816.T+2.79e5T2 1 + 816.T + 2.79e5T^{2}
29 1+1.19e3iT−7.07e5T2 1 + 1.19e3iT - 7.07e5T^{2}
31 1−361.iT−9.23e5T2 1 - 361. iT - 9.23e5T^{2}
37 1+0.0209T+1.87e6T2 1 + 0.0209T + 1.87e6T^{2}
41 1+872.T+2.82e6T2 1 + 872.T + 2.82e6T^{2}
43 1−3.12e3iT−3.41e6T2 1 - 3.12e3iT - 3.41e6T^{2}
47 1−2.65e3T+4.87e6T2 1 - 2.65e3T + 4.87e6T^{2}
53 1−2.95e3T+7.89e6T2 1 - 2.95e3T + 7.89e6T^{2}
59 1−5.41e3T+1.21e7T2 1 - 5.41e3T + 1.21e7T^{2}
61 1−2.25e3iT−1.38e7T2 1 - 2.25e3iT - 1.38e7T^{2}
67 1+2.35e3iT−2.01e7T2 1 + 2.35e3iT - 2.01e7T^{2}
71 1+660.iT−2.54e7T2 1 + 660. iT - 2.54e7T^{2}
73 1−5.94e3iT−2.83e7T2 1 - 5.94e3iT - 2.83e7T^{2}
79 1+7.62e3iT−3.89e7T2 1 + 7.62e3iT - 3.89e7T^{2}
83 1−3.52e3iT−4.74e7T2 1 - 3.52e3iT - 4.74e7T^{2}
89 1+5.94e3T+6.27e7T2 1 + 5.94e3T + 6.27e7T^{2}
97 1+679.iT−8.85e7T2 1 + 679. iT - 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

−9.799589447081457836973796677823, −8.501917300122027481791811640678, −7.957126268849100945609803418268, −7.06652645074044052589272144522, −6.23092200932113509965026582400, −5.73259152544991928297435781411, −4.11467491492820987468543540673, −2.84118608705108982408317028783, −2.07174179901394854900230796956, −0.64097840774818177837666306991, 0.20738417081511124396963897595, 2.35178131783266834034525928849, 3.46979225077749585688355338518, 4.05107687208905419134367995484, 5.25584481077056159030884270627, 5.92846509055166384871821868962, 7.02269350124486191402654546240, 8.300609489188615820037045582374, 8.951572349482876397431820680518, 9.933882436174443296075240362601

Graph of the ZZ-function along the critical line