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Properties

Label 2-8640-1.1-c1-0-62
Degree 22
Conductor 86408640
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 6·13-s + 7·17-s + 7·19-s − 7·23-s + 25-s − 6·29-s − 3·31-s + 2·35-s + 6·37-s + 4·41-s + 8·43-s + 4·47-s − 3·49-s + 5·53-s + 6·59-s + 3·61-s + 6·65-s − 10·67-s − 12·71-s + 16·73-s − 79-s + 9·83-s + 7·85-s − 4·89-s + 12·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 1.66·13-s + 1.69·17-s + 1.60·19-s − 1.45·23-s + 1/5·25-s − 1.11·29-s − 0.538·31-s + 0.338·35-s + 0.986·37-s + 0.624·41-s + 1.21·43-s + 0.583·47-s − 3/7·49-s + 0.686·53-s + 0.781·59-s + 0.384·61-s + 0.744·65-s − 1.22·67-s − 1.42·71-s + 1.87·73-s − 0.112·79-s + 0.987·83-s + 0.759·85-s − 0.423·89-s + 1.25·91-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: 11
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: 00
Selberg data: (2, 8640, ( :1/2), 1)(2,\ 8640,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2465668093.246566809
L(12)L(\frac12) ≈\approx 3.2465668093.246566809
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 1−6T+pT2 1 - 6 T + p T^{2}
17 1−7T+pT2 1 - 7 T + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1+7T+pT2 1 + 7 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−4T+pT2 1 - 4 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 1−3T+pT2 1 - 3 T + p T^{2}
67 1+10T+pT2 1 + 10 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 1−9T+pT2 1 - 9 T + p T^{2}
89 1+4T+pT2 1 + 4 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

−7.81239295473233379077784529073, −7.28348193923657612111973074087, −6.13804575160525039712766109347, −5.69568468995998131868803278612, −5.26148488082895507630327656550, −4.05297786308371571243287583478, −3.62349445900645266899467343816, −2.62933892402557776457814827284, −1.54270187960776649729784848340, −1.00305233293312481201857581927, 1.00305233293312481201857581927, 1.54270187960776649729784848340, 2.62933892402557776457814827284, 3.62349445900645266899467343816, 4.05297786308371571243287583478, 5.26148488082895507630327656550, 5.69568468995998131868803278612, 6.13804575160525039712766109347, 7.28348193923657612111973074087, 7.81239295473233379077784529073

Graph of the ZZ-function along the critical line