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Properties

Label 2-285-1.1-c9-0-14
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  − 6.41·2-s − 81·3-s − 470.·4-s + 625·5-s + 519.·6-s − 7.66e3·7-s + 6.30e3·8-s + 6.56e3·9-s − 4.00e3·10-s + 3.71e3·11-s + 3.81e4·12-s + 1.40e5·13-s + 4.91e4·14-s − 5.06e4·15-s + 2.00e5·16-s − 2.94e4·17-s − 4.20e4·18-s − 1.30e5·19-s − 2.94e5·20-s + 6.20e5·21-s − 2.38e4·22-s + 5.43e5·23-s − 5.10e5·24-s + 3.90e5·25-s − 8.97e5·26-s − 5.31e5·27-s + 3.60e6·28-s + ⋯
L(s)  = 1  − 0.283·2-s − 0.577·3-s − 0.919·4-s + 0.447·5-s + 0.163·6-s − 1.20·7-s + 0.544·8-s + 0.333·9-s − 0.126·10-s + 0.0765·11-s + 0.530·12-s + 1.35·13-s + 0.341·14-s − 0.258·15-s + 0.765·16-s − 0.0854·17-s − 0.0944·18-s − 0.229·19-s − 0.411·20-s + 0.696·21-s − 0.0216·22-s + 0.404·23-s − 0.314·24-s + 0.200·25-s − 0.385·26-s − 0.192·27-s + 1.10·28-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 0.82949479670.8294947967
L(12)L(\frac12) ≈\approx 0.82949479670.8294947967
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+6.41T+512T2 1 + 6.41T + 512T^{2}
7 1+7.66e3T+4.03e7T2 1 + 7.66e3T + 4.03e7T^{2}
11 1−3.71e3T+2.35e9T2 1 - 3.71e3T + 2.35e9T^{2}
13 1−1.40e5T+1.06e10T2 1 - 1.40e5T + 1.06e10T^{2}
17 1+2.94e4T+1.18e11T2 1 + 2.94e4T + 1.18e11T^{2}
23 1−5.43e5T+1.80e12T2 1 - 5.43e5T + 1.80e12T^{2}
29 1−6.53e5T+1.45e13T2 1 - 6.53e5T + 1.45e13T^{2}
31 1+6.20e6T+2.64e13T2 1 + 6.20e6T + 2.64e13T^{2}
37 1+1.43e7T+1.29e14T2 1 + 1.43e7T + 1.29e14T^{2}
41 1+1.40e7T+3.27e14T2 1 + 1.40e7T + 3.27e14T^{2}
43 1−3.38e7T+5.02e14T2 1 - 3.38e7T + 5.02e14T^{2}
47 1−4.14e7T+1.11e15T2 1 - 4.14e7T + 1.11e15T^{2}
53 1+6.45e7T+3.29e15T2 1 + 6.45e7T + 3.29e15T^{2}
59 1−7.94e7T+8.66e15T2 1 - 7.94e7T + 8.66e15T^{2}
61 1−3.29e7T+1.16e16T2 1 - 3.29e7T + 1.16e16T^{2}
67 1+1.96e8T+2.72e16T2 1 + 1.96e8T + 2.72e16T^{2}
71 1+3.62e8T+4.58e16T2 1 + 3.62e8T + 4.58e16T^{2}
73 1+2.73e8T+5.88e16T2 1 + 2.73e8T + 5.88e16T^{2}
79 1+2.95e8T+1.19e17T2 1 + 2.95e8T + 1.19e17T^{2}
83 1−2.00e8T+1.86e17T2 1 - 2.00e8T + 1.86e17T^{2}
89 1−1.00e9T+3.50e17T2 1 - 1.00e9T + 3.50e17T^{2}
97 1+1.03e9T+7.60e17T2 1 + 1.03e9T + 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

−10.24420256842622118014229201573, −9.242270727346554066318303738907, −8.705127510610751171542761257888, −7.26505326812248557010605115188, −6.21012025284651789196901280147, −5.47762940496209556894276104519, −4.18947024293864303287246607980, −3.25709185327697896925250750672, −1.52101642616700767367612613966, −0.46497635303538942817085906856, 0.46497635303538942817085906856, 1.52101642616700767367612613966, 3.25709185327697896925250750672, 4.18947024293864303287246607980, 5.47762940496209556894276104519, 6.21012025284651789196901280147, 7.26505326812248557010605115188, 8.705127510610751171542761257888, 9.242270727346554066318303738907, 10.24420256842622118014229201573

Graph of the ZZ-function along the critical line