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Properties

Label 2-1080-1.1-c3-0-47
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 + 24.0·7-s + 2.95·11-s − 22.1·13-s − 76.0·17-s − 72.1·19-s − 176.·23-s + 25·25-s − 42.5·29-s − 327.·31-s + 120.·35-s + 182.·37-s − 154.·41-s + 173.·43-s − 338.·47-s + 236.·49-s + 26.5·53-s + 14.7·55-s + 391.·59-s − 191.·61-s − 110.·65-s − 507.·67-s + 576.·71-s − 390.·73-s + 71.2·77-s + 1.22e3·79-s − 247.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.30·7-s + 0.0811·11-s − 0.473·13-s − 1.08·17-s − 0.870·19-s − 1.59·23-s + 0.200·25-s − 0.272·29-s − 1.89·31-s + 0.581·35-s + 0.809·37-s − 0.588·41-s + 0.615·43-s − 1.04·47-s + 0.690·49-s + 0.0688·53-s + 0.0362·55-s + 0.864·59-s − 0.401·61-s − 0.211·65-s − 0.925·67-s + 0.963·71-s − 0.625·73-s + 0.105·77-s + 1.73·79-s − 0.327·83-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 1−24.0T+343T2 1 - 24.0T + 343T^{2}
11 1−2.95T+1.33e3T2 1 - 2.95T + 1.33e3T^{2}
13 1+22.1T+2.19e3T2 1 + 22.1T + 2.19e3T^{2}
17 1+76.0T+4.91e3T2 1 + 76.0T + 4.91e3T^{2}
19 1+72.1T+6.85e3T2 1 + 72.1T + 6.85e3T^{2}
23 1+176.T+1.21e4T2 1 + 176.T + 1.21e4T^{2}
29 1+42.5T+2.43e4T2 1 + 42.5T + 2.43e4T^{2}
31 1+327.T+2.97e4T2 1 + 327.T + 2.97e4T^{2}
37 1−182.T+5.06e4T2 1 - 182.T + 5.06e4T^{2}
41 1+154.T+6.89e4T2 1 + 154.T + 6.89e4T^{2}
43 1−173.T+7.95e4T2 1 - 173.T + 7.95e4T^{2}
47 1+338.T+1.03e5T2 1 + 338.T + 1.03e5T^{2}
53 1−26.5T+1.48e5T2 1 - 26.5T + 1.48e5T^{2}
59 1−391.T+2.05e5T2 1 - 391.T + 2.05e5T^{2}
61 1+191.T+2.26e5T2 1 + 191.T + 2.26e5T^{2}
67 1+507.T+3.00e5T2 1 + 507.T + 3.00e5T^{2}
71 1−576.T+3.57e5T2 1 - 576.T + 3.57e5T^{2}
73 1+390.T+3.89e5T2 1 + 390.T + 3.89e5T^{2}
79 1−1.22e3T+4.93e5T2 1 - 1.22e3T + 4.93e5T^{2}
83 1+247.T+5.71e5T2 1 + 247.T + 5.71e5T^{2}
89 1+1.50e3T+7.04e5T2 1 + 1.50e3T + 7.04e5T^{2}
97 1−959.T+9.12e5T2 1 - 959.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

−9.035664958008461470423023596572, −8.270344110527583338924910666038, −7.52345318262806028737014518240, −6.52251549683500906326761494322, −5.60735474978539539723798262400, −4.71962788640532675026696984438, −3.92601991362238562594339022162, −2.30172847567971195882660939865, −1.69285731331916044775571755629, 0, 1.69285731331916044775571755629, 2.30172847567971195882660939865, 3.92601991362238562594339022162, 4.71962788640532675026696984438, 5.60735474978539539723798262400, 6.52251549683500906326761494322, 7.52345318262806028737014518240, 8.270344110527583338924910666038, 9.035664958008461470423023596572

Graph of the ZZ-function along the critical line