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Properties

Label 2-200-40.29-c5-0-40
Degree 22
Conductor 200200
Sign 0.986−0.162i0.986 - 0.162i
Analytic cond. 32.076732.0767
Root an. cond. 5.663635.66363
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.74 − 5.37i)2-s + 25.2·3-s + (−25.8 − 18.8i)4-s + (44.2 − 135. i)6-s + 185. i·7-s + (−146. + 106. i)8-s + 395.·9-s + 574. i·11-s + (−653. − 475. i)12-s + 65.8·13-s + (996. + 324. i)14-s + (315. + 974. i)16-s + 1.96e3i·17-s + (691. − 2.12e3i)18-s + 611. i·19-s + ⋯
L(s)  = 1  + (0.309 − 0.950i)2-s + 1.62·3-s + (−0.808 − 0.588i)4-s + (0.501 − 1.54i)6-s + 1.42i·7-s + (−0.809 + 0.587i)8-s + 1.62·9-s + 1.43i·11-s + (−1.31 − 0.953i)12-s + 0.108·13-s + (1.35 + 0.441i)14-s + (0.307 + 0.951i)16-s + 1.65i·17-s + (0.503 − 1.54i)18-s + 0.388i·19-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=((0.986−0.162i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+5/2) L(s)=((0.986−0.162i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: 0.986−0.162i0.986 - 0.162i
Analytic conductor: 32.076732.0767
Root analytic conductor: 5.663635.66363
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ200(149,⋅)\chi_{200} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 200, ( :5/2), 0.986−0.162i)(2,\ 200,\ (\ :5/2),\ 0.986 - 0.162i)

Particular Values

L(3)L(3) ≈\approx 3.4978048993.497804899
L(12)L(\frac12) ≈\approx 3.4978048993.497804899
L(72)L(\frac{7}{2}) 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+(−1.74+5.37i)T 1 + (-1.74 + 5.37i)T
5 1 1
good3 1−25.2T+243T2 1 - 25.2T + 243T^{2}
7 1−185.iT−1.68e4T2 1 - 185. iT - 1.68e4T^{2}
11 1−574.iT−1.61e5T2 1 - 574. iT - 1.61e5T^{2}
13 1−65.8T+3.71e5T2 1 - 65.8T + 3.71e5T^{2}
17 1−1.96e3iT−1.41e6T2 1 - 1.96e3iT - 1.41e6T^{2}
19 1−611.iT−2.47e6T2 1 - 611. iT - 2.47e6T^{2}
23 1+2.97e3iT−6.43e6T2 1 + 2.97e3iT - 6.43e6T^{2}
29 1+4.47e3iT−2.05e7T2 1 + 4.47e3iT - 2.05e7T^{2}
31 1−7.72e3T+2.86e7T2 1 - 7.72e3T + 2.86e7T^{2}
37 1+7.28e3T+6.93e7T2 1 + 7.28e3T + 6.93e7T^{2}
41 1+6.22e3T+1.15e8T2 1 + 6.22e3T + 1.15e8T^{2}
43 1+1.43e4T+1.47e8T2 1 + 1.43e4T + 1.47e8T^{2}
47 1−5.91e3iT−2.29e8T2 1 - 5.91e3iT - 2.29e8T^{2}
53 1−2.06e4T+4.18e8T2 1 - 2.06e4T + 4.18e8T^{2}
59 1+1.24e4iT−7.14e8T2 1 + 1.24e4iT - 7.14e8T^{2}
61 1+1.98e4iT−8.44e8T2 1 + 1.98e4iT - 8.44e8T^{2}
67 1−5.62e4T+1.35e9T2 1 - 5.62e4T + 1.35e9T^{2}
71 1+5.57e4T+1.80e9T2 1 + 5.57e4T + 1.80e9T^{2}
73 1+3.79e3iT−2.07e9T2 1 + 3.79e3iT - 2.07e9T^{2}
79 1−7.40e4T+3.07e9T2 1 - 7.40e4T + 3.07e9T^{2}
83 1−1.17e5T+3.93e9T2 1 - 1.17e5T + 3.93e9T^{2}
89 1−2.81e4T+5.58e9T2 1 - 2.81e4T + 5.58e9T^{2}
97 1−3.89e4iT−8.58e9T2 1 - 3.89e4iT - 8.58e9T^{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

−11.99101844585291883577638892275, −10.32706161904429318524041890249, −9.670113865997665811749066936242, −8.651364245424109643631983541680, −8.165990347948864355467582653616, −6.31580245522099844779452226328, −4.76439916900224227427283728387, −3.64112106610417110508069542603, −2.39825166664391482985037810403, −1.85392554923913396798296868579, 0.796336946686190108701030226472, 3.07533728115853232899482053389, 3.72498973378529069129259579317, 5.04908833230467166748975692856, 6.78954369425500017572197797618, 7.50626901821373586014936085428, 8.401024585847149342881336398717, 9.178527058111054720622550393708, 10.20645887505308577982058498091, 11.67857067066579992468232810335

Graph of the ZZ-function along the critical line