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Properties

Label 2-800-40.19-c4-0-39
Degree 22
Conductor 800800
Sign 0.708+0.705i0.708 + 0.705i
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  − 0.715i·3-s − 25.0·7-s + 80.4·9-s + 156.·11-s − 301.·13-s − 123. i·17-s + 322.·19-s + 17.9i·21-s − 45.9·23-s − 115. i·27-s + 1.08e3i·29-s − 1.62e3i·31-s − 112. i·33-s + 895.·37-s + 215. i·39-s + ⋯
L(s)  = 1  − 0.0795i·3-s − 0.512·7-s + 0.993·9-s + 1.29·11-s − 1.78·13-s − 0.427i·17-s + 0.892·19-s + 0.0407i·21-s − 0.0867·23-s − 0.158i·27-s + 1.28i·29-s − 1.69i·31-s − 0.102i·33-s + 0.653·37-s + 0.141i·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.708+0.705i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+2) L(s)=((0.708+0.705i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.708+0.705i0.708 + 0.705i
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), 0.708+0.705i)(2,\ 800,\ (\ :2),\ 0.708 + 0.705i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.0346889472.034688947
L(12)L(\frac12) ≈\approx 2.0346889472.034688947
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+0.715iT−81T2 1 + 0.715iT - 81T^{2}
7 1+25.0T+2.40e3T2 1 + 25.0T + 2.40e3T^{2}
11 1−156.T+1.46e4T2 1 - 156.T + 1.46e4T^{2}
13 1+301.T+2.85e4T2 1 + 301.T + 2.85e4T^{2}
17 1+123.iT−8.35e4T2 1 + 123. iT - 8.35e4T^{2}
19 1−322.T+1.30e5T2 1 - 322.T + 1.30e5T^{2}
23 1+45.9T+2.79e5T2 1 + 45.9T + 2.79e5T^{2}
29 1−1.08e3iT−7.07e5T2 1 - 1.08e3iT - 7.07e5T^{2}
31 1+1.62e3iT−9.23e5T2 1 + 1.62e3iT - 9.23e5T^{2}
37 1−895.T+1.87e6T2 1 - 895.T + 1.87e6T^{2}
41 1+1.09e3T+2.82e6T2 1 + 1.09e3T + 2.82e6T^{2}
43 1−950.iT−3.41e6T2 1 - 950. iT - 3.41e6T^{2}
47 1−1.34e3T+4.87e6T2 1 - 1.34e3T + 4.87e6T^{2}
53 1+709.T+7.89e6T2 1 + 709.T + 7.89e6T^{2}
59 1−4.46e3T+1.21e7T2 1 - 4.46e3T + 1.21e7T^{2}
61 1−933.iT−1.38e7T2 1 - 933. iT - 1.38e7T^{2}
67 1−4.01e3iT−2.01e7T2 1 - 4.01e3iT - 2.01e7T^{2}
71 1−3.55e3iT−2.54e7T2 1 - 3.55e3iT - 2.54e7T^{2}
73 1+5.29e3iT−2.83e7T2 1 + 5.29e3iT - 2.83e7T^{2}
79 1+1.14e4iT−3.89e7T2 1 + 1.14e4iT - 3.89e7T^{2}
83 1+6.58e3iT−4.74e7T2 1 + 6.58e3iT - 4.74e7T^{2}
89 1−6.28e3T+6.27e7T2 1 - 6.28e3T + 6.27e7T^{2}
97 1+1.49e4iT−8.85e7T2 1 + 1.49e4iT - 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

−9.707899746686926118435890858557, −8.978473012318413468621799929281, −7.54981710185149405398129821460, −7.13992860498343973061974818362, −6.21713238581939435189442469016, −5.00772284176586408764795866956, −4.17019143576564062745228996059, −3.05424301968779521143965222112, −1.81282084019503347503785846276, −0.58380119148057025799174220237, 0.906379640922925604765481294656, 2.11864254438893077523348528485, 3.41980841816687339471706027548, 4.34583874277681516729355595229, 5.26167498202264146752330379555, 6.54908155763620292019067944906, 7.07026288552170538815832849392, 7.996675253700443955113106551766, 9.250735943107217486552412427903, 9.733612844737845676560858383991

Graph of the ZZ-function along the critical line