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Properties

Label 2-8640-1.1-c1-0-98
Degree 22
Conductor 86408640
Sign −1-1
Analytic cond. 68.990768.9907
Root an. cond. 8.306068.30606
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 − 2·7-s − 2·13-s + 3·17-s + 5·19-s + 3·23-s + 25-s − 6·29-s − 5·31-s − 2·35-s − 2·37-s − 12·41-s + 8·43-s − 12·47-s − 3·49-s − 3·53-s − 6·59-s + 7·61-s − 2·65-s + 2·67-s + 12·71-s − 16·73-s + 79-s + 15·83-s + 3·85-s + 12·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s + 0.727·17-s + 1.14·19-s + 0.625·23-s + 1/5·25-s − 1.11·29-s − 0.898·31-s − 0.338·35-s − 0.328·37-s − 1.87·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.412·53-s − 0.781·59-s + 0.896·61-s − 0.248·65-s + 0.244·67-s + 1.42·71-s − 1.87·73-s + 0.112·79-s + 1.64·83-s + 0.325·85-s + 1.27·89-s + 0.419·91-s + ⋯

Functional equation

Λ(s)=(8640s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8640s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 86408640    =    26⋅33⋅52^{6} \cdot 3^{3} \cdot 5
Sign: −1-1
Analytic conductor: 68.990768.9907
Root analytic conductor: 8.306068.30606
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8640, ( :1/2), −1)(2,\ 8640,\ (\ :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
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+16T+pT2 1 + 16 T + p T^{2}
79 1−T+pT2 1 - T + p T^{2}
83 1−15T+pT2 1 - 15 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+16T+pT2 1 + 16 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

−7.37682638552384433932572566594, −6.78087153782208371351120932155, −6.04677031648612258852720624225, −5.30452740982438204115491211417, −4.86816953691102107806545887845, −3.53504414022398929649170033593, −3.27283099980780212398513252150, −2.20481643293850327409860751663, −1.27808640296439589403710050675, 0, 1.27808640296439589403710050675, 2.20481643293850327409860751663, 3.27283099980780212398513252150, 3.53504414022398929649170033593, 4.86816953691102107806545887845, 5.30452740982438204115491211417, 6.04677031648612258852720624225, 6.78087153782208371351120932155, 7.37682638552384433932572566594

Graph of the ZZ-function along the critical line