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Properties

Label 2-350-1.1-c5-0-36
Degree 22
Conductor 350350
Sign −1-1
Analytic cond. 56.134356.1343
Root an. cond. 7.492287.49228
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  − 4·2-s + 11·3-s + 16·4-s − 44·6-s − 49·7-s − 64·8-s − 122·9-s − 267·11-s + 176·12-s + 1.08e3·13-s + 196·14-s + 256·16-s + 513·17-s + 488·18-s − 802·19-s − 539·21-s + 1.06e3·22-s + 1.29e3·23-s − 704·24-s − 4.34e3·26-s − 4.01e3·27-s − 784·28-s + 1.77e3·29-s − 2.58e3·31-s − 1.02e3·32-s − 2.93e3·33-s − 2.05e3·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.705·3-s + 1/2·4-s − 0.498·6-s − 0.377·7-s − 0.353·8-s − 0.502·9-s − 0.665·11-s + 0.352·12-s + 1.78·13-s + 0.267·14-s + 1/4·16-s + 0.430·17-s + 0.355·18-s − 0.509·19-s − 0.266·21-s + 0.470·22-s + 0.508·23-s − 0.249·24-s − 1.26·26-s − 1.05·27-s − 0.188·28-s + 0.392·29-s − 0.482·31-s − 0.176·32-s − 0.469·33-s − 0.304·34-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 56.134356.1343
Root analytic conductor: 7.492287.49228
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 350, ( :5/2), −1)(2,\ 350,\ (\ :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)
bad2 1+p2T 1 + p^{2} T
5 1 1
7 1+p2T 1 + p^{2} T
good3 1−11T+p5T2 1 - 11 T + p^{5} T^{2}
11 1+267T+p5T2 1 + 267 T + p^{5} T^{2}
13 1−1087T+p5T2 1 - 1087 T + p^{5} T^{2}
17 1−513T+p5T2 1 - 513 T + p^{5} T^{2}
19 1+802T+p5T2 1 + 802 T + p^{5} T^{2}
23 1−1290T+p5T2 1 - 1290 T + p^{5} T^{2}
29 1−1779T+p5T2 1 - 1779 T + p^{5} T^{2}
31 1+2584T+p5T2 1 + 2584 T + p^{5} T^{2}
37 1+13862T+p5T2 1 + 13862 T + p^{5} T^{2}
41 1+11904T+p5T2 1 + 11904 T + p^{5} T^{2}
43 1−598T+p5T2 1 - 598 T + p^{5} T^{2}
47 1−17019T+p5T2 1 - 17019 T + p^{5} T^{2}
53 1+27852T+p5T2 1 + 27852 T + p^{5} T^{2}
59 1−30912T+p5T2 1 - 30912 T + p^{5} T^{2}
61 1+1780T+p5T2 1 + 1780 T + p^{5} T^{2}
67 1+25052T+p5T2 1 + 25052 T + p^{5} T^{2}
71 1+51984T+p5T2 1 + 51984 T + p^{5} T^{2}
73 1+47690T+p5T2 1 + 47690 T + p^{5} T^{2}
79 1+102121T+p5T2 1 + 102121 T + p^{5} T^{2}
83 1−83676T+p5T2 1 - 83676 T + p^{5} T^{2}
89 1+32400T+p5T2 1 + 32400 T + p^{5} T^{2}
97 1−148645T+p5T2 1 - 148645 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

−10.20242003240162580897755457699, −8.909594847363771695660584004193, −8.607945881738578257862504015549, −7.61416245657015220848157960762, −6.45964758154176663310586217606, −5.48348114496744528615004206462, −3.69565242178110963316303699540, −2.83837887333414565772330317588, −1.49489331137436378747435832505, 0, 1.49489331137436378747435832505, 2.83837887333414565772330317588, 3.69565242178110963316303699540, 5.48348114496744528615004206462, 6.45964758154176663310586217606, 7.61416245657015220848157960762, 8.607945881738578257862504015549, 8.909594847363771695660584004193, 10.20242003240162580897755457699

Graph of the ZZ-function along the critical line