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Properties

Label 2-912-1.1-c1-0-7
Degree 22
Conductor 912912
Sign 11
Analytic cond. 7.282357.28235
Root an. cond. 2.698582.69858
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 6·17-s + 19-s + 4·23-s − 25-s + 27-s − 2·29-s − 4·31-s + 4·33-s + 10·37-s + 2·39-s + 10·41-s − 4·43-s + 2·45-s + 4·47-s − 7·49-s − 6·51-s − 10·53-s + 8·55-s + 57-s − 12·59-s + 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.229·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 0.583·47-s − 49-s − 0.840·51-s − 1.37·53-s + 1.07·55-s + 0.132·57-s − 1.56·59-s + 1.79·61-s + ⋯

Functional equation

Λ(s)=(912s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(912s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 912912    =    24⋅3⋅192^{4} \cdot 3 \cdot 19
Sign: 11
Analytic conductor: 7.282357.28235
Root analytic conductor: 2.698582.69858
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 912, ( :1/2), 1)(2,\ 912,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3884724342.388472434
L(12)L(\frac12) ≈\approx 2.3884724342.388472434
L(32)L(\frac{3}{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 1
3 1−T 1 - T
19 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p 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

−9.786869452003372550002449800084, −9.259741991745774642651687902060, −8.686112233403729699672924385749, −7.54920325839787930805373575681, −6.57061735757055000239038218298, −5.95928356394221873345303919204, −4.65672229415455481414222119791, −3.72816659383691276241941645312, −2.47746893642927435624182028712, −1.41228664440352179477085709942, 1.41228664440352179477085709942, 2.47746893642927435624182028712, 3.72816659383691276241941645312, 4.65672229415455481414222119791, 5.95928356394221873345303919204, 6.57061735757055000239038218298, 7.54920325839787930805373575681, 8.686112233403729699672924385749, 9.259741991745774642651687902060, 9.786869452003372550002449800084

Graph of the ZZ-function along the critical line