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Properties

Label 2-285-1.1-c9-0-15
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  − 1.75·2-s − 81·3-s − 508.·4-s + 625·5-s + 141.·6-s + 2.35e3·7-s + 1.78e3·8-s + 6.56e3·9-s − 1.09e3·10-s − 5.80e4·11-s + 4.12e4·12-s − 3.56e4·13-s − 4.12e3·14-s − 5.06e4·15-s + 2.57e5·16-s + 5.67e5·17-s − 1.14e4·18-s − 1.30e5·19-s − 3.18e5·20-s − 1.90e5·21-s + 1.01e5·22-s − 1.17e6·23-s − 1.44e5·24-s + 3.90e5·25-s + 6.24e4·26-s − 5.31e5·27-s − 1.19e6·28-s + ⋯
L(s)  = 1  − 0.0774·2-s − 0.577·3-s − 0.994·4-s + 0.447·5-s + 0.0446·6-s + 0.370·7-s + 0.154·8-s + 0.333·9-s − 0.0346·10-s − 1.19·11-s + 0.573·12-s − 0.345·13-s − 0.0287·14-s − 0.258·15-s + 0.982·16-s + 1.64·17-s − 0.0258·18-s − 0.229·19-s − 0.444·20-s − 0.214·21-s + 0.0925·22-s − 0.876·23-s − 0.0891·24-s + 0.200·25-s + 0.0267·26-s − 0.192·27-s − 0.368·28-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 1.0069878041.006987804
L(12)L(\frac12) ≈\approx 1.0069878041.006987804
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+1.75T+512T2 1 + 1.75T + 512T^{2}
7 1−2.35e3T+4.03e7T2 1 - 2.35e3T + 4.03e7T^{2}
11 1+5.80e4T+2.35e9T2 1 + 5.80e4T + 2.35e9T^{2}
13 1+3.56e4T+1.06e10T2 1 + 3.56e4T + 1.06e10T^{2}
17 1−5.67e5T+1.18e11T2 1 - 5.67e5T + 1.18e11T^{2}
23 1+1.17e6T+1.80e12T2 1 + 1.17e6T + 1.80e12T^{2}
29 1+1.92e6T+1.45e13T2 1 + 1.92e6T + 1.45e13T^{2}
31 1−2.50e6T+2.64e13T2 1 - 2.50e6T + 2.64e13T^{2}
37 1−8.97e6T+1.29e14T2 1 - 8.97e6T + 1.29e14T^{2}
41 1−1.69e6T+3.27e14T2 1 - 1.69e6T + 3.27e14T^{2}
43 1+9.56e6T+5.02e14T2 1 + 9.56e6T + 5.02e14T^{2}
47 1−6.17e6T+1.11e15T2 1 - 6.17e6T + 1.11e15T^{2}
53 1+7.25e7T+3.29e15T2 1 + 7.25e7T + 3.29e15T^{2}
59 1+1.64e8T+8.66e15T2 1 + 1.64e8T + 8.66e15T^{2}
61 1+9.49e7T+1.16e16T2 1 + 9.49e7T + 1.16e16T^{2}
67 1−1.74e8T+2.72e16T2 1 - 1.74e8T + 2.72e16T^{2}
71 1−1.58e8T+4.58e16T2 1 - 1.58e8T + 4.58e16T^{2}
73 1+5.12e6T+5.88e16T2 1 + 5.12e6T + 5.88e16T^{2}
79 1+2.90e8T+1.19e17T2 1 + 2.90e8T + 1.19e17T^{2}
83 1−2.26e8T+1.86e17T2 1 - 2.26e8T + 1.86e17T^{2}
89 1+8.41e8T+3.50e17T2 1 + 8.41e8T + 3.50e17T^{2}
97 1+1.02e9T+7.60e17T2 1 + 1.02e9T + 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

−10.07562433337516815632495116233, −9.597965498595450030858799832089, −8.198008341287422152649406619178, −7.62975528640918878870892162563, −6.02187895079481946793027209525, −5.28739911493237841148809055025, −4.49751822542694385919178560191, −3.12820374553830946050281983283, −1.64193662530629540187482100417, −0.48186099471475849484586717465, 0.48186099471475849484586717465, 1.64193662530629540187482100417, 3.12820374553830946050281983283, 4.49751822542694385919178560191, 5.28739911493237841148809055025, 6.02187895079481946793027209525, 7.62975528640918878870892162563, 8.198008341287422152649406619178, 9.597965498595450030858799832089, 10.07562433337516815632495116233

Graph of the ZZ-function along the critical line