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Properties

Label 2-798-1.1-c3-0-21
Degree 22
Conductor 798798
Sign 11
Analytic cond. 47.083547.0835
Root an. cond. 6.861746.86174
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s + 7.25·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 14.5·10-s + 6.99·11-s − 12·12-s − 6.86·13-s + 14·14-s − 21.7·15-s + 16·16-s + 43.1·17-s + 18·18-s − 19·19-s + 29.0·20-s − 21·21-s + 13.9·22-s + 198.·23-s − 24·24-s − 72.3·25-s − 13.7·26-s − 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.649·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.459·10-s + 0.191·11-s − 0.288·12-s − 0.146·13-s + 0.267·14-s − 0.374·15-s + 0.250·16-s + 0.615·17-s + 0.235·18-s − 0.229·19-s + 0.324·20-s − 0.218·21-s + 0.135·22-s + 1.80·23-s − 0.204·24-s − 0.578·25-s − 0.103·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(798s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(798s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 798798    =    2⋅3⋅7⋅192 \cdot 3 \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 47.083547.0835
Root analytic conductor: 6.861746.86174
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 798, ( :3/2), 1)(2,\ 798,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.4096070513.409607051
L(12)L(\frac12) ≈\approx 3.4096070513.409607051
L(52)L(\frac{5}{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 1−2T 1 - 2T
3 1+3T 1 + 3T
7 1−7T 1 - 7T
19 1+19T 1 + 19T
good5 1−7.25T+125T2 1 - 7.25T + 125T^{2}
11 1−6.99T+1.33e3T2 1 - 6.99T + 1.33e3T^{2}
13 1+6.86T+2.19e3T2 1 + 6.86T + 2.19e3T^{2}
17 1−43.1T+4.91e3T2 1 - 43.1T + 4.91e3T^{2}
23 1−198.T+1.21e4T2 1 - 198.T + 1.21e4T^{2}
29 1+29.1T+2.43e4T2 1 + 29.1T + 2.43e4T^{2}
31 1−39.1T+2.97e4T2 1 - 39.1T + 2.97e4T^{2}
37 1−152.T+5.06e4T2 1 - 152.T + 5.06e4T^{2}
41 1−94.7T+6.89e4T2 1 - 94.7T + 6.89e4T^{2}
43 1+101.T+7.95e4T2 1 + 101.T + 7.95e4T^{2}
47 1+386.T+1.03e5T2 1 + 386.T + 1.03e5T^{2}
53 1−310.T+1.48e5T2 1 - 310.T + 1.48e5T^{2}
59 1−207.T+2.05e5T2 1 - 207.T + 2.05e5T^{2}
61 1−266.T+2.26e5T2 1 - 266.T + 2.26e5T^{2}
67 1+31.7T+3.00e5T2 1 + 31.7T + 3.00e5T^{2}
71 1+280.T+3.57e5T2 1 + 280.T + 3.57e5T^{2}
73 1−219.T+3.89e5T2 1 - 219.T + 3.89e5T^{2}
79 1−1.18e3T+4.93e5T2 1 - 1.18e3T + 4.93e5T^{2}
83 1−1.07e3T+5.71e5T2 1 - 1.07e3T + 5.71e5T^{2}
89 1−1.59e3T+7.04e5T2 1 - 1.59e3T + 7.04e5T^{2}
97 1+975.T+9.12e5T2 1 + 975.T + 9.12e5T^{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

−10.02159863679279011689978661647, −9.194525498489806474381475814585, −7.997393523414776526566671573691, −7.03533062747303681813241794800, −6.21581599676082251273819167046, −5.37134943086596578576044386797, −4.69295911333345613633247435834, −3.48039673816754829414151966094, −2.20965914515537405905342072335, −1.01378992586530547788583438467, 1.01378992586530547788583438467, 2.20965914515537405905342072335, 3.48039673816754829414151966094, 4.69295911333345613633247435834, 5.37134943086596578576044386797, 6.21581599676082251273819167046, 7.03533062747303681813241794800, 7.997393523414776526566671573691, 9.194525498489806474381475814585, 10.02159863679279011689978661647

Graph of the ZZ-function along the critical line