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Properties

Label 2-1440-1.1-c3-0-51
Degree 22
Conductor 14401440
Sign −1-1
Analytic cond. 84.962784.9627
Root an. cond. 9.217529.21752
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 + 10.3·7-s − 57.3·11-s − 22.3·13-s + 106.·17-s − 43.7·19-s + 16.4·23-s + 25·25-s + 57.5·29-s + 175.·31-s + 51.6·35-s − 280.·37-s − 157.·41-s − 457.·43-s − 18.6·47-s − 236.·49-s − 370.·53-s − 286.·55-s − 416.·59-s + 867.·61-s − 111.·65-s − 37.8·67-s + 83.6·71-s − 12.1·73-s − 592.·77-s − 175.·79-s − 572.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.558·7-s − 1.57·11-s − 0.475·13-s + 1.51·17-s − 0.528·19-s + 0.149·23-s + 0.200·25-s + 0.368·29-s + 1.01·31-s + 0.249·35-s − 1.24·37-s − 0.600·41-s − 1.62·43-s − 0.0578·47-s − 0.688·49-s − 0.960·53-s − 0.703·55-s − 0.918·59-s + 1.82·61-s − 0.212·65-s − 0.0690·67-s + 0.139·71-s − 0.0195·73-s − 0.877·77-s − 0.249·79-s − 0.756·83-s + ⋯

Functional equation

Λ(s)=(1440s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1440s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14401440    =    25⋅32⋅52^{5} \cdot 3^{2} \cdot 5
Sign: −1-1
Analytic conductor: 84.962784.9627
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1440, ( :3/2), −1)(2,\ 1440,\ (\ :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 1−10.3T+343T2 1 - 10.3T + 343T^{2}
11 1+57.3T+1.33e3T2 1 + 57.3T + 1.33e3T^{2}
13 1+22.3T+2.19e3T2 1 + 22.3T + 2.19e3T^{2}
17 1−106.T+4.91e3T2 1 - 106.T + 4.91e3T^{2}
19 1+43.7T+6.85e3T2 1 + 43.7T + 6.85e3T^{2}
23 1−16.4T+1.21e4T2 1 - 16.4T + 1.21e4T^{2}
29 1−57.5T+2.43e4T2 1 - 57.5T + 2.43e4T^{2}
31 1−175.T+2.97e4T2 1 - 175.T + 2.97e4T^{2}
37 1+280.T+5.06e4T2 1 + 280.T + 5.06e4T^{2}
41 1+157.T+6.89e4T2 1 + 157.T + 6.89e4T^{2}
43 1+457.T+7.95e4T2 1 + 457.T + 7.95e4T^{2}
47 1+18.6T+1.03e5T2 1 + 18.6T + 1.03e5T^{2}
53 1+370.T+1.48e5T2 1 + 370.T + 1.48e5T^{2}
59 1+416.T+2.05e5T2 1 + 416.T + 2.05e5T^{2}
61 1−867.T+2.26e5T2 1 - 867.T + 2.26e5T^{2}
67 1+37.8T+3.00e5T2 1 + 37.8T + 3.00e5T^{2}
71 1−83.6T+3.57e5T2 1 - 83.6T + 3.57e5T^{2}
73 1+12.1T+3.89e5T2 1 + 12.1T + 3.89e5T^{2}
79 1+175.T+4.93e5T2 1 + 175.T + 4.93e5T^{2}
83 1+572.T+5.71e5T2 1 + 572.T + 5.71e5T^{2}
89 1−622.T+7.04e5T2 1 - 622.T + 7.04e5T^{2}
97 1+1.21e3T+9.12e5T2 1 + 1.21e3T + 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

−8.530035457335660671347475402042, −8.061573897460417842583104908360, −7.24586793390817502779695517051, −6.24006443379490506803945415717, −5.18932707492727457907114223289, −4.90377693494979912030258726224, −3.39496139058713302190351582471, −2.50291738472753237903198838970, −1.42677167007891290358703017084, 0, 1.42677167007891290358703017084, 2.50291738472753237903198838970, 3.39496139058713302190351582471, 4.90377693494979912030258726224, 5.18932707492727457907114223289, 6.24006443379490506803945415717, 7.24586793390817502779695517051, 8.061573897460417842583104908360, 8.530035457335660671347475402042

Graph of the ZZ-function along the critical line