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Properties

Label 2-147-1.1-c5-0-18
Degree 22
Conductor 147147
Sign −1-1
Analytic cond. 23.576423.5764
Root an. cond. 4.855554.85555
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 9·3-s + 4·4-s − 78·5-s − 54·6-s + 168·8-s + 81·9-s + 468·10-s + 444·11-s + 36·12-s + 442·13-s − 702·15-s − 1.13e3·16-s + 126·17-s − 486·18-s − 2.68e3·19-s − 312·20-s − 2.66e3·22-s + 4.20e3·23-s + 1.51e3·24-s + 2.95e3·25-s − 2.65e3·26-s + 729·27-s − 5.44e3·29-s + 4.21e3·30-s − 80·31-s + 1.44e3·32-s + ⋯
L(s)  = 1  − 1.06·2-s + 0.577·3-s + 1/8·4-s − 1.39·5-s − 0.612·6-s + 0.928·8-s + 1/3·9-s + 1.47·10-s + 1.10·11-s + 0.0721·12-s + 0.725·13-s − 0.805·15-s − 1.10·16-s + 0.105·17-s − 0.353·18-s − 1.70·19-s − 0.174·20-s − 1.17·22-s + 1.65·23-s + 0.535·24-s + 0.946·25-s − 0.769·26-s + 0.192·27-s − 1.20·29-s + 0.854·30-s − 0.0149·31-s + 0.248·32-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: −1-1
Analytic conductor: 23.576423.5764
Root analytic conductor: 4.855554.85555
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 147, ( :5/2), −1)(2,\ 147,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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−p2T 1 - p^{2} T
7 1 1
good2 1+3pT+p5T2 1 + 3 p T + p^{5} T^{2}
5 1+78T+p5T2 1 + 78 T + p^{5} T^{2}
11 1−444T+p5T2 1 - 444 T + p^{5} T^{2}
13 1−34pT+p5T2 1 - 34 p T + p^{5} T^{2}
17 1−126T+p5T2 1 - 126 T + p^{5} T^{2}
19 1+2684T+p5T2 1 + 2684 T + p^{5} T^{2}
23 1−4200T+p5T2 1 - 4200 T + p^{5} T^{2}
29 1+5442T+p5T2 1 + 5442 T + p^{5} T^{2}
31 1+80T+p5T2 1 + 80 T + p^{5} T^{2}
37 1+5434T+p5T2 1 + 5434 T + p^{5} T^{2}
41 1+7962T+p5T2 1 + 7962 T + p^{5} T^{2}
43 1+268pT+p5T2 1 + 268 p T + p^{5} T^{2}
47 1−13920T+p5T2 1 - 13920 T + p^{5} T^{2}
53 1+9594T+p5T2 1 + 9594 T + p^{5} T^{2}
59 1+27492T+p5T2 1 + 27492 T + p^{5} T^{2}
61 1+49478T+p5T2 1 + 49478 T + p^{5} T^{2}
67 1+59356T+p5T2 1 + 59356 T + p^{5} T^{2}
71 1−32040T+p5T2 1 - 32040 T + p^{5} T^{2}
73 1−61846T+p5T2 1 - 61846 T + p^{5} T^{2}
79 1+65776T+p5T2 1 + 65776 T + p^{5} T^{2}
83 1+40188T+p5T2 1 + 40188 T + p^{5} T^{2}
89 1−7974T+p5T2 1 - 7974 T + p^{5} T^{2}
97 1−143662T+p5T2 1 - 143662 T + p^{5} T^{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

−11.33691382444602661676796984865, −10.60096114619047474880460011031, −9.120639165452258990306953998291, −8.663423568224121446368503890948, −7.70532215906419380097637815362, −6.73864989978780196308920997253, −4.48494206107780944893104157030, −3.56105570507868716757861496148, −1.45458680556475004975894606522, 0, 1.45458680556475004975894606522, 3.56105570507868716757861496148, 4.48494206107780944893104157030, 6.73864989978780196308920997253, 7.70532215906419380097637815362, 8.663423568224121446368503890948, 9.120639165452258990306953998291, 10.60096114619047474880460011031, 11.33691382444602661676796984865

Graph of the ZZ-function along the critical line