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Properties

Label 2-800-40.19-c4-0-63
Degree 22
Conductor 800800
Sign −0.263+0.964i-0.263 + 0.964i
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  + 4.51i·3-s + 50.0·7-s + 60.5·9-s − 158.·11-s − 97.3·13-s − 285. i·17-s + 414.·19-s + 226. i·21-s − 546.·23-s + 639. i·27-s − 1.16e3i·29-s − 369. i·31-s − 717. i·33-s − 1.42e3·37-s − 439. i·39-s + ⋯
L(s)  = 1  + 0.502i·3-s + 1.02·7-s + 0.747·9-s − 1.31·11-s − 0.575·13-s − 0.987i·17-s + 1.14·19-s + 0.513i·21-s − 1.03·23-s + 0.877i·27-s − 1.38i·29-s − 0.384i·31-s − 0.659i·33-s − 1.03·37-s − 0.289i·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((−0.263+0.964i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+2) L(s)=((−0.263+0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: −0.263+0.964i-0.263 + 0.964i
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), −0.263+0.964i)(2,\ 800,\ (\ :2),\ -0.263 + 0.964i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.0735594341.073559434
L(12)L(\frac12) ≈\approx 1.0735594341.073559434
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 1−4.51iT−81T2 1 - 4.51iT - 81T^{2}
7 1−50.0T+2.40e3T2 1 - 50.0T + 2.40e3T^{2}
11 1+158.T+1.46e4T2 1 + 158.T + 1.46e4T^{2}
13 1+97.3T+2.85e4T2 1 + 97.3T + 2.85e4T^{2}
17 1+285.iT−8.35e4T2 1 + 285. iT - 8.35e4T^{2}
19 1−414.T+1.30e5T2 1 - 414.T + 1.30e5T^{2}
23 1+546.T+2.79e5T2 1 + 546.T + 2.79e5T^{2}
29 1+1.16e3iT−7.07e5T2 1 + 1.16e3iT - 7.07e5T^{2}
31 1+369.iT−9.23e5T2 1 + 369. iT - 9.23e5T^{2}
37 1+1.42e3T+1.87e6T2 1 + 1.42e3T + 1.87e6T^{2}
41 1+2.90e3T+2.82e6T2 1 + 2.90e3T + 2.82e6T^{2}
43 1+905.iT−3.41e6T2 1 + 905. iT - 3.41e6T^{2}
47 1−3.02e3T+4.87e6T2 1 - 3.02e3T + 4.87e6T^{2}
53 1+1.60e3T+7.89e6T2 1 + 1.60e3T + 7.89e6T^{2}
59 1+1.80e3T+1.21e7T2 1 + 1.80e3T + 1.21e7T^{2}
61 1−1.11e3iT−1.38e7T2 1 - 1.11e3iT - 1.38e7T^{2}
67 1−3.91e3iT−2.01e7T2 1 - 3.91e3iT - 2.01e7T^{2}
71 1+4.22e3iT−2.54e7T2 1 + 4.22e3iT - 2.54e7T^{2}
73 1−4.66e3iT−2.83e7T2 1 - 4.66e3iT - 2.83e7T^{2}
79 1−6.81e3iT−3.89e7T2 1 - 6.81e3iT - 3.89e7T^{2}
83 1−3.16e3iT−4.74e7T2 1 - 3.16e3iT - 4.74e7T^{2}
89 1+1.07e4T+6.27e7T2 1 + 1.07e4T + 6.27e7T^{2}
97 1+1.00e4iT−8.85e7T2 1 + 1.00e4iT - 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

−9.792178774196842762276395959239, −8.513351148543799289588739448176, −7.68122847046051658503390714262, −7.13425977524284615492806740972, −5.55377918599067146687433900157, −4.98401655014809427996923453441, −4.13623177884822519228932403586, −2.81612516517230272188179903067, −1.71907466850914227694306956022, −0.23304485324746792940006185561, 1.31701818057558217435292253035, 2.10635030105684518549194147346, 3.45481697960151547367368764415, 4.77414676907637923434852670119, 5.34817386145970927657252516251, 6.59415137360480149977711809689, 7.62988824828070185166440467417, 7.911340229185042509321855714186, 8.972516538610908828454755737322, 10.25223112476033840010487416643

Graph of the ZZ-function along the critical line