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Properties

Label 2-1080-1.1-c3-0-45
Degree 22
Conductor 10801080
Sign −1-1
Analytic cond. 63.722063.7220
Root an. cond. 7.982617.98261
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 11.6·7-s − 47.4·11-s + 84.2·13-s − 26.9·17-s − 152.·19-s − 177.·23-s + 25·25-s + 60.0·29-s − 92.1·31-s + 58.1·35-s − 221.·37-s + 115.·41-s + 383.·43-s + 317.·47-s − 207.·49-s − 257.·53-s − 237.·55-s − 642.·59-s − 662.·61-s + 421.·65-s + 597.·67-s − 500.·71-s + 989.·73-s − 552.·77-s − 517.·79-s − 605.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.627·7-s − 1.30·11-s + 1.79·13-s − 0.384·17-s − 1.83·19-s − 1.60·23-s + 0.200·25-s + 0.384·29-s − 0.533·31-s + 0.280·35-s − 0.985·37-s + 0.438·41-s + 1.36·43-s + 0.984·47-s − 0.605·49-s − 0.667·53-s − 0.582·55-s − 1.41·59-s − 1.38·61-s + 0.804·65-s + 1.08·67-s − 0.836·71-s + 1.58·73-s − 0.817·77-s − 0.737·79-s − 0.801·83-s + ⋯

Functional equation

Λ(s)=(1080s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1080s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 63.722063.7220
Root analytic conductor: 7.982617.98261
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1080, ( :3/2), −1)(2,\ 1080,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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 1−5T 1 - 5T
good7 1−11.6T+343T2 1 - 11.6T + 343T^{2}
11 1+47.4T+1.33e3T2 1 + 47.4T + 1.33e3T^{2}
13 1−84.2T+2.19e3T2 1 - 84.2T + 2.19e3T^{2}
17 1+26.9T+4.91e3T2 1 + 26.9T + 4.91e3T^{2}
19 1+152.T+6.85e3T2 1 + 152.T + 6.85e3T^{2}
23 1+177.T+1.21e4T2 1 + 177.T + 1.21e4T^{2}
29 1−60.0T+2.43e4T2 1 - 60.0T + 2.43e4T^{2}
31 1+92.1T+2.97e4T2 1 + 92.1T + 2.97e4T^{2}
37 1+221.T+5.06e4T2 1 + 221.T + 5.06e4T^{2}
41 1−115.T+6.89e4T2 1 - 115.T + 6.89e4T^{2}
43 1−383.T+7.95e4T2 1 - 383.T + 7.95e4T^{2}
47 1−317.T+1.03e5T2 1 - 317.T + 1.03e5T^{2}
53 1+257.T+1.48e5T2 1 + 257.T + 1.48e5T^{2}
59 1+642.T+2.05e5T2 1 + 642.T + 2.05e5T^{2}
61 1+662.T+2.26e5T2 1 + 662.T + 2.26e5T^{2}
67 1−597.T+3.00e5T2 1 - 597.T + 3.00e5T^{2}
71 1+500.T+3.57e5T2 1 + 500.T + 3.57e5T^{2}
73 1−989.T+3.89e5T2 1 - 989.T + 3.89e5T^{2}
79 1+517.T+4.93e5T2 1 + 517.T + 4.93e5T^{2}
83 1+605.T+5.71e5T2 1 + 605.T + 5.71e5T^{2}
89 1+1.51e3T+7.04e5T2 1 + 1.51e3T + 7.04e5T^{2}
97 1−742.T+9.12e5T2 1 - 742.T + 9.12e5T^{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

−8.855298369238887270457962807610, −8.368788313502090377075880932202, −7.58631022200336632697029538722, −6.28511906109972511700367694204, −5.84508362651120888937787002222, −4.69339257953013455333637222275, −3.82862202276236538551606746883, −2.45817011220563662751963284146, −1.57940059891343308903227160333, 0, 1.57940059891343308903227160333, 2.45817011220563662751963284146, 3.82862202276236538551606746883, 4.69339257953013455333637222275, 5.84508362651120888937787002222, 6.28511906109972511700367694204, 7.58631022200336632697029538722, 8.368788313502090377075880932202, 8.855298369238887270457962807610

Graph of the ZZ-function along the critical line