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Properties

Label 2-147-1.1-c5-0-11
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  + 5.44·2-s − 9·3-s − 2.33·4-s + 36·5-s − 49.0·6-s − 187.·8-s + 81·9-s + 196.·10-s + 184.·11-s + 21.0·12-s + 147.·13-s − 324·15-s − 943.·16-s + 1.96e3·17-s + 441.·18-s + 1.89e3·19-s − 84.1·20-s + 1.00e3·22-s + 136.·23-s + 1.68e3·24-s − 1.82e3·25-s + 805.·26-s − 729·27-s − 1.25e3·29-s − 1.76e3·30-s + 8.96e3·31-s + 844.·32-s + ⋯
L(s)  = 1  + 0.962·2-s − 0.577·3-s − 0.0730·4-s + 0.643·5-s − 0.555·6-s − 1.03·8-s + 0.333·9-s + 0.620·10-s + 0.459·11-s + 0.0421·12-s + 0.242·13-s − 0.371·15-s − 0.921·16-s + 1.65·17-s + 0.320·18-s + 1.20·19-s − 0.0470·20-s + 0.442·22-s + 0.0539·23-s + 0.596·24-s − 0.585·25-s + 0.233·26-s − 0.192·27-s − 0.278·29-s − 0.357·30-s + 1.67·31-s + 0.145·32-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 2.6986602002.698660200
L(12)L(\frac12) ≈\approx 2.6986602002.698660200
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 1−5.44T+32T2 1 - 5.44T + 32T^{2}
5 1−36T+3.12e3T2 1 - 36T + 3.12e3T^{2}
11 1−184.T+1.61e5T2 1 - 184.T + 1.61e5T^{2}
13 1−147.T+3.71e5T2 1 - 147.T + 3.71e5T^{2}
17 1−1.96e3T+1.41e6T2 1 - 1.96e3T + 1.41e6T^{2}
19 1−1.89e3T+2.47e6T2 1 - 1.89e3T + 2.47e6T^{2}
23 1−136.T+6.43e6T2 1 - 136.T + 6.43e6T^{2}
29 1+1.25e3T+2.05e7T2 1 + 1.25e3T + 2.05e7T^{2}
31 1−8.96e3T+2.86e7T2 1 - 8.96e3T + 2.86e7T^{2}
37 1−1.28e4T+6.93e7T2 1 - 1.28e4T + 6.93e7T^{2}
41 1−8.97e3T+1.15e8T2 1 - 8.97e3T + 1.15e8T^{2}
43 1−1.35e4T+1.47e8T2 1 - 1.35e4T + 1.47e8T^{2}
47 1+2.00e4T+2.29e8T2 1 + 2.00e4T + 2.29e8T^{2}
53 1−9.33e3T+4.18e8T2 1 - 9.33e3T + 4.18e8T^{2}
59 1−8.86e3T+7.14e8T2 1 - 8.86e3T + 7.14e8T^{2}
61 1+4.11e4T+8.44e8T2 1 + 4.11e4T + 8.44e8T^{2}
67 1+5.53e4T+1.35e9T2 1 + 5.53e4T + 1.35e9T^{2}
71 1+6.38e4T+1.80e9T2 1 + 6.38e4T + 1.80e9T^{2}
73 1−4.12e4T+2.07e9T2 1 - 4.12e4T + 2.07e9T^{2}
79 1−1.69e4T+3.07e9T2 1 - 1.69e4T + 3.07e9T^{2}
83 1+1.01e5T+3.93e9T2 1 + 1.01e5T + 3.93e9T^{2}
89 1−8.71e4T+5.58e9T2 1 - 8.71e4T + 5.58e9T^{2}
97 1+1.18e5T+8.58e9T2 1 + 1.18e5T + 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

−12.19289985383651485720853556257, −11.54671236992937161041205407819, −10.04104152857849535960409107043, −9.323694854004559267706260818217, −7.73888855441177545298502215249, −6.17308660688735500903120277548, −5.58783053482362562003933596407, −4.39315914171940491356431612961, −3.06159944040201632589000446740, −1.04175397356406014468170844194, 1.04175397356406014468170844194, 3.06159944040201632589000446740, 4.39315914171940491356431612961, 5.58783053482362562003933596407, 6.17308660688735500903120277548, 7.73888855441177545298502215249, 9.323694854004559267706260818217, 10.04104152857849535960409107043, 11.54671236992937161041205407819, 12.19289985383651485720853556257

Graph of the ZZ-function along the critical line