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Properties

Label 2-560-1.1-c5-0-40
Degree 22
Conductor 560560
Sign −1-1
Analytic cond. 89.814989.8149
Root an. cond. 9.477079.47707
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.24·3-s − 25·5-s − 49·7-s − 203.·9-s − 89.4·11-s + 459.·13-s + 156.·15-s + 701.·17-s + 396.·19-s + 306.·21-s + 4.31e3·23-s + 625·25-s + 2.79e3·27-s − 1.43e3·29-s − 3.66e3·31-s + 558.·33-s + 1.22e3·35-s − 3.16e3·37-s − 2.86e3·39-s − 6.35e3·41-s + 1.55e4·43-s + 5.09e3·45-s + 3.99e3·47-s + 2.40e3·49-s − 4.38e3·51-s − 2.44e4·53-s + 2.23e3·55-s + ⋯
L(s)  = 1  − 0.400·3-s − 0.447·5-s − 0.377·7-s − 0.839·9-s − 0.222·11-s + 0.753·13-s + 0.179·15-s + 0.588·17-s + 0.251·19-s + 0.151·21-s + 1.70·23-s + 0.200·25-s + 0.737·27-s − 0.317·29-s − 0.685·31-s + 0.0893·33-s + 0.169·35-s − 0.380·37-s − 0.302·39-s − 0.590·41-s + 1.27·43-s + 0.375·45-s + 0.263·47-s + 0.142·49-s − 0.235·51-s − 1.19·53-s + 0.0996·55-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 89.814989.8149
Root analytic conductor: 9.477079.47707
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 560, ( :5/2), −1)(2,\ 560,\ (\ :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
5 1+25T 1 + 25T
7 1+49T 1 + 49T
good3 1+6.24T+243T2 1 + 6.24T + 243T^{2}
11 1+89.4T+1.61e5T2 1 + 89.4T + 1.61e5T^{2}
13 1−459.T+3.71e5T2 1 - 459.T + 3.71e5T^{2}
17 1−701.T+1.41e6T2 1 - 701.T + 1.41e6T^{2}
19 1−396.T+2.47e6T2 1 - 396.T + 2.47e6T^{2}
23 1−4.31e3T+6.43e6T2 1 - 4.31e3T + 6.43e6T^{2}
29 1+1.43e3T+2.05e7T2 1 + 1.43e3T + 2.05e7T^{2}
31 1+3.66e3T+2.86e7T2 1 + 3.66e3T + 2.86e7T^{2}
37 1+3.16e3T+6.93e7T2 1 + 3.16e3T + 6.93e7T^{2}
41 1+6.35e3T+1.15e8T2 1 + 6.35e3T + 1.15e8T^{2}
43 1−1.55e4T+1.47e8T2 1 - 1.55e4T + 1.47e8T^{2}
47 1−3.99e3T+2.29e8T2 1 - 3.99e3T + 2.29e8T^{2}
53 1+2.44e4T+4.18e8T2 1 + 2.44e4T + 4.18e8T^{2}
59 1−2.39e3T+7.14e8T2 1 - 2.39e3T + 7.14e8T^{2}
61 1+2.10e4T+8.44e8T2 1 + 2.10e4T + 8.44e8T^{2}
67 1+853.T+1.35e9T2 1 + 853.T + 1.35e9T^{2}
71 1−1.37e3T+1.80e9T2 1 - 1.37e3T + 1.80e9T^{2}
73 1−1.22e4T+2.07e9T2 1 - 1.22e4T + 2.07e9T^{2}
79 1−5.10e4T+3.07e9T2 1 - 5.10e4T + 3.07e9T^{2}
83 1+1.76e4T+3.93e9T2 1 + 1.76e4T + 3.93e9T^{2}
89 1+1.83e4T+5.58e9T2 1 + 1.83e4T + 5.58e9T^{2}
97 1+1.03e5T+8.58e9T2 1 + 1.03e5T + 8.58e9T^{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

−9.437545750775655891767029553858, −8.705181032959064670184164425361, −7.74413190142952824025696488517, −6.78251490105059860401189540159, −5.80237491974682400045490602115, −5.01225265515751654584688826594, −3.65798524363005277916745848626, −2.82655552320397336928600724721, −1.13744468595766172569024831670, 0, 1.13744468595766172569024831670, 2.82655552320397336928600724721, 3.65798524363005277916745848626, 5.01225265515751654584688826594, 5.80237491974682400045490602115, 6.78251490105059860401189540159, 7.74413190142952824025696488517, 8.705181032959064670184164425361, 9.437545750775655891767029553858

Graph of the ZZ-function along the critical line