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Properties

Label 2-8640-1.1-c1-0-81
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 − 4·7-s − 2·11-s − 4·13-s + 17-s + 5·19-s + 5·23-s + 25-s − 8·29-s + 7·31-s − 4·35-s + 6·37-s + 6·41-s + 2·43-s + 8·47-s + 9·49-s − 9·53-s − 2·55-s − 4·59-s − 13·61-s − 4·65-s + 10·67-s − 6·71-s − 6·73-s + 8·77-s + 9·79-s + 17·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 0.603·11-s − 1.10·13-s + 0.242·17-s + 1.14·19-s + 1.04·23-s + 1/5·25-s − 1.48·29-s + 1.25·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s + 0.304·43-s + 1.16·47-s + 9/7·49-s − 1.23·53-s − 0.269·55-s − 0.520·59-s − 1.66·61-s − 0.496·65-s + 1.22·67-s − 0.712·71-s − 0.702·73-s + 0.911·77-s + 1.01·79-s + 1.86·83-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+4T+pT2 1 + 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−T+pT2 1 - T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1−5T+pT2 1 - 5 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+13T+pT2 1 + 13 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−9T+pT2 1 - 9 T + p T^{2}
83 1−17T+pT2 1 - 17 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−7.49021375169655068173068034665, −6.71401739160618710686470724247, −6.05954716596321482019902208949, −5.42016231751570044953010087520, −4.74472917965118669238450357257, −3.74003194570339284765478358434, −2.85015981870912461136487956707, −2.56805366284799917081919448512, −1.12737072596111806526309478815, 0, 1.12737072596111806526309478815, 2.56805366284799917081919448512, 2.85015981870912461136487956707, 3.74003194570339284765478358434, 4.74472917965118669238450357257, 5.42016231751570044953010087520, 6.05954716596321482019902208949, 6.71401739160618710686470724247, 7.49021375169655068173068034665

Graph of the ZZ-function along the critical line