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Properties

Label 2-285-1.1-c9-0-47
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  + 9.10·2-s − 81·3-s − 429.·4-s + 625·5-s − 737.·6-s + 6.71e3·7-s − 8.56e3·8-s + 6.56e3·9-s + 5.68e3·10-s + 7.38e4·11-s + 3.47e4·12-s + 4.82e4·13-s + 6.11e4·14-s − 5.06e4·15-s + 1.41e5·16-s + 3.02e5·17-s + 5.97e4·18-s − 1.30e5·19-s − 2.68e5·20-s − 5.43e5·21-s + 6.72e5·22-s + 2.35e6·23-s + 6.93e5·24-s + 3.90e5·25-s + 4.38e5·26-s − 5.31e5·27-s − 2.88e6·28-s + ⋯
L(s)  = 1  + 0.402·2-s − 0.577·3-s − 0.838·4-s + 0.447·5-s − 0.232·6-s + 1.05·7-s − 0.739·8-s + 0.333·9-s + 0.179·10-s + 1.52·11-s + 0.483·12-s + 0.468·13-s + 0.425·14-s − 0.258·15-s + 0.540·16-s + 0.877·17-s + 0.134·18-s − 0.229·19-s − 0.374·20-s − 0.610·21-s + 0.611·22-s + 1.75·23-s + 0.426·24-s + 0.200·25-s + 0.188·26-s − 0.192·27-s − 0.885·28-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 2.8908830922.890883092
L(12)L(\frac12) ≈\approx 2.8908830922.890883092
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 1−9.10T+512T2 1 - 9.10T + 512T^{2}
7 1−6.71e3T+4.03e7T2 1 - 6.71e3T + 4.03e7T^{2}
11 1−7.38e4T+2.35e9T2 1 - 7.38e4T + 2.35e9T^{2}
13 1−4.82e4T+1.06e10T2 1 - 4.82e4T + 1.06e10T^{2}
17 1−3.02e5T+1.18e11T2 1 - 3.02e5T + 1.18e11T^{2}
23 1−2.35e6T+1.80e12T2 1 - 2.35e6T + 1.80e12T^{2}
29 1+2.23e6T+1.45e13T2 1 + 2.23e6T + 1.45e13T^{2}
31 1−1.87e5T+2.64e13T2 1 - 1.87e5T + 2.64e13T^{2}
37 1+4.75e6T+1.29e14T2 1 + 4.75e6T + 1.29e14T^{2}
41 1+9.93e6T+3.27e14T2 1 + 9.93e6T + 3.27e14T^{2}
43 1+6.79e6T+5.02e14T2 1 + 6.79e6T + 5.02e14T^{2}
47 1−2.32e7T+1.11e15T2 1 - 2.32e7T + 1.11e15T^{2}
53 1−6.12e7T+3.29e15T2 1 - 6.12e7T + 3.29e15T^{2}
59 1−1.19e7T+8.66e15T2 1 - 1.19e7T + 8.66e15T^{2}
61 1−1.12e4T+1.16e16T2 1 - 1.12e4T + 1.16e16T^{2}
67 1+1.64e8T+2.72e16T2 1 + 1.64e8T + 2.72e16T^{2}
71 1+2.51e8T+4.58e16T2 1 + 2.51e8T + 4.58e16T^{2}
73 1+1.11e8T+5.88e16T2 1 + 1.11e8T + 5.88e16T^{2}
79 1−3.07e8T+1.19e17T2 1 - 3.07e8T + 1.19e17T^{2}
83 1+2.22e8T+1.86e17T2 1 + 2.22e8T + 1.86e17T^{2}
89 1+3.96e8T+3.50e17T2 1 + 3.96e8T + 3.50e17T^{2}
97 1−1.27e9T+7.60e17T2 1 - 1.27e9T + 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

−10.29298907841265842845413460014, −9.196698895472659940372242563672, −8.585829198020997893291791125462, −7.21315310107207929948630882173, −6.06633058662662722983880622504, −5.23139116087686483720141526393, −4.39980707991666001276822078537, −3.38075638818063455821643929124, −1.55651129063327280765132020012, −0.833014284840587501281655905950, 0.833014284840587501281655905950, 1.55651129063327280765132020012, 3.38075638818063455821643929124, 4.39980707991666001276822078537, 5.23139116087686483720141526393, 6.06633058662662722983880622504, 7.21315310107207929948630882173, 8.585829198020997893291791125462, 9.196698895472659940372242563672, 10.29298907841265842845413460014

Graph of the ZZ-function along the critical line