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Properties

Label 2-43560-1.1-c1-0-30
Degree 22
Conductor 4356043560
Sign −1-1
Analytic cond. 347.828347.828
Root an. cond. 18.650118.6501
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s + 2·13-s − 3·17-s − 8·19-s − 23-s + 25-s − 2·29-s − 3·31-s + 4·35-s − 4·37-s − 6·41-s + 10·43-s + 9·47-s + 9·49-s + 5·53-s + 6·59-s + 61-s − 2·65-s + 4·71-s + 4·73-s + 79-s + 8·83-s + 3·85-s − 8·91-s + 8·95-s − 12·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 0.554·13-s − 0.727·17-s − 1.83·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s − 0.538·31-s + 0.676·35-s − 0.657·37-s − 0.937·41-s + 1.52·43-s + 1.31·47-s + 9/7·49-s + 0.686·53-s + 0.781·59-s + 0.128·61-s − 0.248·65-s + 0.474·71-s + 0.468·73-s + 0.112·79-s + 0.878·83-s + 0.325·85-s − 0.838·91-s + 0.820·95-s − 1.21·97-s + ⋯

Functional equation

Λ(s)=(43560s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(43560s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4356043560    =    23⋅32⋅5⋅1122^{3} \cdot 3^{2} \cdot 5 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 347.828347.828
Root analytic conductor: 18.650118.6501
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 43560, ( :1/2), −1)(2,\ 43560,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1
5 1+T 1 + T
11 1 1
good7 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+T+pT2 1 + T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1−5T+pT2 1 - 5 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−T+pT2 1 - T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+12T+pT2 1 + 12 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

−15.13132580529677, −14.49710961252488, −13.68730393300735, −13.43052749429125, −12.76669691859014, −12.49943136146045, −11.99889804707527, −11.09481318098967, −10.86254055080419, −10.26657949474715, −9.728326403427427, −8.956099564719584, −8.796247918626607, −8.139480130635492, −7.305013321289507, −6.874719889618268, −6.338718018226614, −5.908241865215614, −5.175453298366482, −4.198614310496342, −3.945924920581084, −3.323248806785887, −2.501210542689044, −1.964112977489065, −0.7137050774097439, 0, 0.7137050774097439, 1.964112977489065, 2.501210542689044, 3.323248806785887, 3.945924920581084, 4.198614310496342, 5.175453298366482, 5.908241865215614, 6.338718018226614, 6.874719889618268, 7.305013321289507, 8.139480130635492, 8.796247918626607, 8.956099564719584, 9.728326403427427, 10.26657949474715, 10.86254055080419, 11.09481318098967, 11.99889804707527, 12.49943136146045, 12.76669691859014, 13.43052749429125, 13.68730393300735, 14.49710961252488, 15.13132580529677

Graph of the ZZ-function along the critical line