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Properties

Label 2-285-1.1-c9-0-45
Degree 22
Conductor 285285
Sign 11
Analytic cond. 146.785146.785
Root an. cond. 12.115412.1154
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 31.2·2-s − 81·3-s + 461.·4-s + 625·5-s + 2.52e3·6-s + 9.73e3·7-s + 1.57e3·8-s + 6.56e3·9-s − 1.95e4·10-s + 2.50e4·11-s − 3.73e4·12-s − 1.29e5·13-s − 3.03e5·14-s − 5.06e4·15-s − 2.85e5·16-s + 5.47e5·17-s − 2.04e5·18-s − 1.30e5·19-s + 2.88e5·20-s − 7.88e5·21-s − 7.80e5·22-s + 1.95e6·23-s − 1.27e5·24-s + 3.90e5·25-s + 4.03e6·26-s − 5.31e5·27-s + 4.49e6·28-s + ⋯
L(s)  = 1  − 1.37·2-s − 0.577·3-s + 0.901·4-s + 0.447·5-s + 0.796·6-s + 1.53·7-s + 0.135·8-s + 0.333·9-s − 0.616·10-s + 0.515·11-s − 0.520·12-s − 1.25·13-s − 2.11·14-s − 0.258·15-s − 1.08·16-s + 1.59·17-s − 0.459·18-s − 0.229·19-s + 0.403·20-s − 0.885·21-s − 0.710·22-s + 1.45·23-s − 0.0784·24-s + 0.200·25-s + 1.73·26-s − 0.192·27-s + 1.38·28-s + ⋯

Functional equation

Λ(s)=(285s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(285s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 285285    =    3⋅5⋅193 \cdot 5 \cdot 19
Sign: 11
Analytic conductor: 146.785146.785
Root analytic conductor: 12.115412.1154
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 285, ( :9/2), 1)(2,\ 285,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 1.4716121151.471612115
L(12)L(\frac12) ≈\approx 1.4716121151.471612115
L(112)L(\frac{11}{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)
bad3 1+81T 1 + 81T
5 1−625T 1 - 625T
19 1+1.30e5T 1 + 1.30e5T
good2 1+31.2T+512T2 1 + 31.2T + 512T^{2}
7 1−9.73e3T+4.03e7T2 1 - 9.73e3T + 4.03e7T^{2}
11 1−2.50e4T+2.35e9T2 1 - 2.50e4T + 2.35e9T^{2}
13 1+1.29e5T+1.06e10T2 1 + 1.29e5T + 1.06e10T^{2}
17 1−5.47e5T+1.18e11T2 1 - 5.47e5T + 1.18e11T^{2}
23 1−1.95e6T+1.80e12T2 1 - 1.95e6T + 1.80e12T^{2}
29 1−6.56e6T+1.45e13T2 1 - 6.56e6T + 1.45e13T^{2}
31 1−3.33e6T+2.64e13T2 1 - 3.33e6T + 2.64e13T^{2}
37 1−1.09e7T+1.29e14T2 1 - 1.09e7T + 1.29e14T^{2}
41 1+1.47e7T+3.27e14T2 1 + 1.47e7T + 3.27e14T^{2}
43 1−3.78e7T+5.02e14T2 1 - 3.78e7T + 5.02e14T^{2}
47 1−5.22e6T+1.11e15T2 1 - 5.22e6T + 1.11e15T^{2}
53 1+1.08e8T+3.29e15T2 1 + 1.08e8T + 3.29e15T^{2}
59 1+5.97e6T+8.66e15T2 1 + 5.97e6T + 8.66e15T^{2}
61 1−1.18e8T+1.16e16T2 1 - 1.18e8T + 1.16e16T^{2}
67 1−2.19e8T+2.72e16T2 1 - 2.19e8T + 2.72e16T^{2}
71 1−1.40e8T+4.58e16T2 1 - 1.40e8T + 4.58e16T^{2}
73 1−3.61e8T+5.88e16T2 1 - 3.61e8T + 5.88e16T^{2}
79 1+5.17e8T+1.19e17T2 1 + 5.17e8T + 1.19e17T^{2}
83 1+7.82e8T+1.86e17T2 1 + 7.82e8T + 1.86e17T^{2}
89 1−8.91e8T+3.50e17T2 1 - 8.91e8T + 3.50e17T^{2}
97 1−3.50e8T+7.60e17T2 1 - 3.50e8T + 7.60e17T^{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

−10.12478668201187124185047071161, −9.425672206658819192003268148573, −8.317352065483546705861315197417, −7.63415617102415600601906679879, −6.68183205383416801547955826275, −5.23909902875497721110518336139, −4.56689686122567564974959238230, −2.52606584126199572505725558054, −1.31795610066982129080658122412, −0.824600655623805676111193837215, 0.824600655623805676111193837215, 1.31795610066982129080658122412, 2.52606584126199572505725558054, 4.56689686122567564974959238230, 5.23909902875497721110518336139, 6.68183205383416801547955826275, 7.63415617102415600601906679879, 8.317352065483546705861315197417, 9.425672206658819192003268148573, 10.12478668201187124185047071161

Graph of the ZZ-function along the critical line