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Properties

Label 2-1080-1.1-c5-0-53
Degree 22
Conductor 10801080
Sign −1-1
Analytic cond. 173.214173.214
Root an. cond. 13.161013.1610
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25·5-s − 234·7-s + 347·11-s − 33·13-s + 237·17-s + 1.49e3·19-s − 2.81e3·23-s + 625·25-s − 5.51e3·29-s + 2.91e3·31-s − 5.85e3·35-s + 5.60e3·37-s + 4.71e3·41-s + 1.04e4·43-s + 5.96e3·47-s + 3.79e4·49-s − 1.79e4·53-s + 8.67e3·55-s + 3.03e4·59-s − 3.55e4·61-s − 825·65-s − 1.24e4·67-s + 7.52e3·71-s + 3.63e4·73-s − 8.11e4·77-s − 2.27e4·79-s + 4.62e4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.80·7-s + 0.864·11-s − 0.0541·13-s + 0.198·17-s + 0.950·19-s − 1.10·23-s + 1/5·25-s − 1.21·29-s + 0.544·31-s − 0.807·35-s + 0.672·37-s + 0.438·41-s + 0.864·43-s + 0.393·47-s + 2.25·49-s − 0.878·53-s + 0.386·55-s + 1.13·59-s − 1.22·61-s − 0.0242·65-s − 0.339·67-s + 0.177·71-s + 0.798·73-s − 1.56·77-s − 0.409·79-s + 0.736·83-s + ⋯

Functional equation

Λ(s)=(1080s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(1080s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10801080    =    23⋅33⋅52^{3} \cdot 3^{3} \cdot 5
Sign: −1-1
Analytic conductor: 173.214173.214
Root analytic conductor: 13.161013.1610
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1080, ( :5/2), −1)(2,\ 1080,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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−p2T 1 - p^{2} T
good7 1+234T+p5T2 1 + 234 T + p^{5} T^{2}
11 1−347T+p5T2 1 - 347 T + p^{5} T^{2}
13 1+33T+p5T2 1 + 33 T + p^{5} T^{2}
17 1−237T+p5T2 1 - 237 T + p^{5} T^{2}
19 1−1496T+p5T2 1 - 1496 T + p^{5} T^{2}
23 1+2811T+p5T2 1 + 2811 T + p^{5} T^{2}
29 1+5513T+p5T2 1 + 5513 T + p^{5} T^{2}
31 1−2911T+p5T2 1 - 2911 T + p^{5} T^{2}
37 1−5602T+p5T2 1 - 5602 T + p^{5} T^{2}
41 1−4716T+p5T2 1 - 4716 T + p^{5} T^{2}
43 1−10479T+p5T2 1 - 10479 T + p^{5} T^{2}
47 1−5963T+p5T2 1 - 5963 T + p^{5} T^{2}
53 1+17964T+p5T2 1 + 17964 T + p^{5} T^{2}
59 1−30372T+p5T2 1 - 30372 T + p^{5} T^{2}
61 1+35530T+p5T2 1 + 35530 T + p^{5} T^{2}
67 1+12476T+p5T2 1 + 12476 T + p^{5} T^{2}
71 1−7520T+p5T2 1 - 7520 T + p^{5} T^{2}
73 1−36378T+p5T2 1 - 36378 T + p^{5} T^{2}
79 1+22727T+p5T2 1 + 22727 T + p^{5} T^{2}
83 1−46254T+p5T2 1 - 46254 T + p^{5} T^{2}
89 1+58832T+p5T2 1 + 58832 T + p^{5} T^{2}
97 1+145906T+p5T2 1 + 145906 T + p^{5} 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

−9.097939515311542447195151765023, −7.81523232578969914992187772138, −6.91896988159313213859541350913, −6.17847617541267733123498598133, −5.60481081737031640810194253204, −4.15429481309921995968474263381, −3.38942587383030071762017549956, −2.44991101315731285001679188035, −1.12714601808032824039813874305, 0, 1.12714601808032824039813874305, 2.44991101315731285001679188035, 3.38942587383030071762017549956, 4.15429481309921995968474263381, 5.60481081737031640810194253204, 6.17847617541267733123498598133, 6.91896988159313213859541350913, 7.81523232578969914992187772138, 9.097939515311542447195151765023

Graph of the ZZ-function along the critical line