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Properties

Label 2-147-1.1-c5-0-15
Degree 22
Conductor 147147
Sign 11
Analytic cond. 23.576423.5764
Root an. cond. 4.855554.85555
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.31·2-s + 9·3-s − 3.71·4-s + 103.·5-s − 47.8·6-s + 189.·8-s + 81·9-s − 550.·10-s + 653.·11-s − 33.4·12-s − 138.·13-s + 931.·15-s − 891.·16-s + 1.17e3·17-s − 430.·18-s − 1.71e3·19-s − 384.·20-s − 3.47e3·22-s − 4.02e3·23-s + 1.70e3·24-s + 7.58e3·25-s + 734.·26-s + 729·27-s + 2.64e3·29-s − 4.95e3·30-s − 2.87e3·31-s − 1.33e3·32-s + ⋯
L(s)  = 1  − 0.940·2-s + 0.577·3-s − 0.116·4-s + 1.85·5-s − 0.542·6-s + 1.04·8-s + 0.333·9-s − 1.74·10-s + 1.62·11-s − 0.0670·12-s − 0.226·13-s + 1.06·15-s − 0.870·16-s + 0.985·17-s − 0.313·18-s − 1.08·19-s − 0.215·20-s − 1.53·22-s − 1.58·23-s + 0.605·24-s + 2.42·25-s + 0.212·26-s + 0.192·27-s + 0.585·29-s − 1.00·30-s − 0.537·31-s − 0.231·32-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: 11
Analytic conductor: 23.576423.5764
Root analytic conductor: 4.855554.85555
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 147, ( :5/2), 1)(2,\ 147,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.0417370972.041737097
L(12)L(\frac12) ≈\approx 2.0417370972.041737097
L(72)L(\frac{7}{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 1−9T 1 - 9T
7 1 1
good2 1+5.31T+32T2 1 + 5.31T + 32T^{2}
5 1−103.T+3.12e3T2 1 - 103.T + 3.12e3T^{2}
11 1−653.T+1.61e5T2 1 - 653.T + 1.61e5T^{2}
13 1+138.T+3.71e5T2 1 + 138.T + 3.71e5T^{2}
17 1−1.17e3T+1.41e6T2 1 - 1.17e3T + 1.41e6T^{2}
19 1+1.71e3T+2.47e6T2 1 + 1.71e3T + 2.47e6T^{2}
23 1+4.02e3T+6.43e6T2 1 + 4.02e3T + 6.43e6T^{2}
29 1−2.64e3T+2.05e7T2 1 - 2.64e3T + 2.05e7T^{2}
31 1+2.87e3T+2.86e7T2 1 + 2.87e3T + 2.86e7T^{2}
37 1−2.85e3T+6.93e7T2 1 - 2.85e3T + 6.93e7T^{2}
41 1−216.T+1.15e8T2 1 - 216.T + 1.15e8T^{2}
43 1−2.92e3T+1.47e8T2 1 - 2.92e3T + 1.47e8T^{2}
47 1−1.48e4T+2.29e8T2 1 - 1.48e4T + 2.29e8T^{2}
53 1−2.11e4T+4.18e8T2 1 - 2.11e4T + 4.18e8T^{2}
59 1−3.46e4T+7.14e8T2 1 - 3.46e4T + 7.14e8T^{2}
61 1+8.75e3T+8.44e8T2 1 + 8.75e3T + 8.44e8T^{2}
67 1+1.20e4T+1.35e9T2 1 + 1.20e4T + 1.35e9T^{2}
71 1+3.55e4T+1.80e9T2 1 + 3.55e4T + 1.80e9T^{2}
73 1−3.34e4T+2.07e9T2 1 - 3.34e4T + 2.07e9T^{2}
79 1−4.31e4T+3.07e9T2 1 - 4.31e4T + 3.07e9T^{2}
83 1+4.33e4T+3.93e9T2 1 + 4.33e4T + 3.93e9T^{2}
89 1+1.03e5T+5.58e9T2 1 + 1.03e5T + 5.58e9T^{2}
97 1+8.62e4T+8.58e9T2 1 + 8.62e4T + 8.58e9T^{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

−12.24021461025367975578381086043, −10.52854891473395275107743391318, −9.801520685781124159301607285382, −9.212669589709511611108954473714, −8.349820473367381836486921658888, −6.87817671375199741233615366752, −5.76759868748204443316294305813, −4.14312150386472702852850793792, −2.15498759034873486040431736349, −1.19969189940168022109989643484, 1.19969189940168022109989643484, 2.15498759034873486040431736349, 4.14312150386472702852850793792, 5.76759868748204443316294305813, 6.87817671375199741233615366752, 8.349820473367381836486921658888, 9.212669589709511611108954473714, 9.801520685781124159301607285382, 10.52854891473395275107743391318, 12.24021461025367975578381086043

Graph of the ZZ-function along the critical line