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Properties

Label 2-147-1.1-c5-0-26
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  + 10.2·2-s + 9·3-s + 72.5·4-s + 23.7·5-s + 92.0·6-s + 414.·8-s + 81·9-s + 242.·10-s + 465.·11-s + 652.·12-s − 1.01e3·13-s + 213.·15-s + 1.91e3·16-s + 561.·17-s + 828.·18-s − 1.38e3·19-s + 1.72e3·20-s + 4.75e3·22-s + 4.11e3·23-s + 3.72e3·24-s − 2.56e3·25-s − 1.04e4·26-s + 729·27-s − 2.38e3·29-s + 2.18e3·30-s + 2.95e3·31-s + 6.32e3·32-s + ⋯
L(s)  = 1  + 1.80·2-s + 0.577·3-s + 2.26·4-s + 0.424·5-s + 1.04·6-s + 2.28·8-s + 0.333·9-s + 0.767·10-s + 1.16·11-s + 1.30·12-s − 1.67·13-s + 0.245·15-s + 1.87·16-s + 0.471·17-s + 0.602·18-s − 0.881·19-s + 0.963·20-s + 2.09·22-s + 1.62·23-s + 1.32·24-s − 0.819·25-s − 3.02·26-s + 0.192·27-s − 0.525·29-s + 0.443·30-s + 0.551·31-s + 1.09·32-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 7.3749191707.374919170
L(12)L(\frac12) ≈\approx 7.3749191707.374919170
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 1−10.2T+32T2 1 - 10.2T + 32T^{2}
5 1−23.7T+3.12e3T2 1 - 23.7T + 3.12e3T^{2}
11 1−465.T+1.61e5T2 1 - 465.T + 1.61e5T^{2}
13 1+1.01e3T+3.71e5T2 1 + 1.01e3T + 3.71e5T^{2}
17 1−561.T+1.41e6T2 1 - 561.T + 1.41e6T^{2}
19 1+1.38e3T+2.47e6T2 1 + 1.38e3T + 2.47e6T^{2}
23 1−4.11e3T+6.43e6T2 1 - 4.11e3T + 6.43e6T^{2}
29 1+2.38e3T+2.05e7T2 1 + 2.38e3T + 2.05e7T^{2}
31 1−2.95e3T+2.86e7T2 1 - 2.95e3T + 2.86e7T^{2}
37 1+9.90e3T+6.93e7T2 1 + 9.90e3T + 6.93e7T^{2}
41 1+4.47e3T+1.15e8T2 1 + 4.47e3T + 1.15e8T^{2}
43 1−5.18e3T+1.47e8T2 1 - 5.18e3T + 1.47e8T^{2}
47 1+3.12e3T+2.29e8T2 1 + 3.12e3T + 2.29e8T^{2}
53 1−1.14e3T+4.18e8T2 1 - 1.14e3T + 4.18e8T^{2}
59 1−2.74e4T+7.14e8T2 1 - 2.74e4T + 7.14e8T^{2}
61 1+2.11e4T+8.44e8T2 1 + 2.11e4T + 8.44e8T^{2}
67 1+5.55e4T+1.35e9T2 1 + 5.55e4T + 1.35e9T^{2}
71 1+6.07e3T+1.80e9T2 1 + 6.07e3T + 1.80e9T^{2}
73 1+1.67e4T+2.07e9T2 1 + 1.67e4T + 2.07e9T^{2}
79 1+4.84e3T+3.07e9T2 1 + 4.84e3T + 3.07e9T^{2}
83 1−6.01e4T+3.93e9T2 1 - 6.01e4T + 3.93e9T^{2}
89 1+6.24e4T+5.58e9T2 1 + 6.24e4T + 5.58e9T^{2}
97 1+6.36e4T+8.58e9T2 1 + 6.36e4T + 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

−12.39635573232361945791735854184, −11.64004192126054526133855338309, −10.30413947452200291864367984914, −9.116183313441131394148809465866, −7.40531694628912075524526028804, −6.54724847647037084352551908362, −5.27006957305223147410689500831, −4.23979870455222121548272321789, −3.02353116087827822112528580437, −1.86141064336625750071593011878, 1.86141064336625750071593011878, 3.02353116087827822112528580437, 4.23979870455222121548272321789, 5.27006957305223147410689500831, 6.54724847647037084352551908362, 7.40531694628912075524526028804, 9.116183313441131394148809465866, 10.30413947452200291864367984914, 11.64004192126054526133855338309, 12.39635573232361945791735854184

Graph of the ZZ-function along the critical line