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Properties

Label 2-560-1.1-c3-0-13
Degree 22
Conductor 560560
Sign 11
Analytic cond. 33.041033.0410
Root an. cond. 5.748135.74813
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.65·3-s − 5·5-s + 7·7-s − 5.31·9-s + 52.2·11-s + 30.6·13-s − 23.2·15-s + 37.2·17-s − 80.2·19-s + 32.5·21-s − 25.8·23-s + 25·25-s − 150.·27-s + 20.9·29-s + 314.·31-s + 243.·33-s − 35·35-s + 197.·37-s + 142.·39-s + 11.3·41-s + 33.8·43-s + 26.5·45-s + 361.·47-s + 49·49-s + 173.·51-s + 153.·53-s − 261.·55-s + ⋯
L(s)  = 1  + 0.896·3-s − 0.447·5-s + 0.377·7-s − 0.196·9-s + 1.43·11-s + 0.654·13-s − 0.400·15-s + 0.531·17-s − 0.968·19-s + 0.338·21-s − 0.234·23-s + 0.200·25-s − 1.07·27-s + 0.134·29-s + 1.82·31-s + 1.28·33-s − 0.169·35-s + 0.875·37-s + 0.586·39-s + 0.0432·41-s + 0.119·43-s + 0.0880·45-s + 1.12·47-s + 0.142·49-s + 0.475·51-s + 0.396·53-s − 0.640·55-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 33.041033.0410
Root analytic conductor: 5.748135.74813
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 560, ( :3/2), 1)(2,\ 560,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.8345979962.834597996
L(12)L(\frac12) ≈\approx 2.8345979962.834597996
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
5 1+5T 1 + 5T
7 1−7T 1 - 7T
good3 1−4.65T+27T2 1 - 4.65T + 27T^{2}
11 1−52.2T+1.33e3T2 1 - 52.2T + 1.33e3T^{2}
13 1−30.6T+2.19e3T2 1 - 30.6T + 2.19e3T^{2}
17 1−37.2T+4.91e3T2 1 - 37.2T + 4.91e3T^{2}
19 1+80.2T+6.85e3T2 1 + 80.2T + 6.85e3T^{2}
23 1+25.8T+1.21e4T2 1 + 25.8T + 1.21e4T^{2}
29 1−20.9T+2.43e4T2 1 - 20.9T + 2.43e4T^{2}
31 1−314.T+2.97e4T2 1 - 314.T + 2.97e4T^{2}
37 1−197.T+5.06e4T2 1 - 197.T + 5.06e4T^{2}
41 1−11.3T+6.89e4T2 1 - 11.3T + 6.89e4T^{2}
43 1−33.8T+7.95e4T2 1 - 33.8T + 7.95e4T^{2}
47 1−361.T+1.03e5T2 1 - 361.T + 1.03e5T^{2}
53 1−153.T+1.48e5T2 1 - 153.T + 1.48e5T^{2}
59 1−616T+2.05e5T2 1 - 616T + 2.05e5T^{2}
61 1−15.2T+2.26e5T2 1 - 15.2T + 2.26e5T^{2}
67 1−166.T+3.00e5T2 1 - 166.T + 3.00e5T^{2}
71 1−952T+3.57e5T2 1 - 952T + 3.57e5T^{2}
73 1+148.T+3.89e5T2 1 + 148.T + 3.89e5T^{2}
79 1+857.T+4.93e5T2 1 + 857.T + 4.93e5T^{2}
83 1+660.T+5.71e5T2 1 + 660.T + 5.71e5T^{2}
89 1+45.7T+7.04e5T2 1 + 45.7T + 7.04e5T^{2}
97 1−1.68e3T+9.12e5T2 1 - 1.68e3T + 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

−10.28106354695442536119364281249, −9.249509787807976425309992784933, −8.524353351140309678324763188754, −7.984567827427055002540320018741, −6.79548820583060812781754778118, −5.87775508944929476062047480217, −4.35857900253479350063769821294, −3.65145473247428242929094068901, −2.43721924732272623395284712248, −1.04039831827976666961139235274, 1.04039831827976666961139235274, 2.43721924732272623395284712248, 3.65145473247428242929094068901, 4.35857900253479350063769821294, 5.87775508944929476062047480217, 6.79548820583060812781754778118, 7.984567827427055002540320018741, 8.524353351140309678324763188754, 9.249509787807976425309992784933, 10.28106354695442536119364281249

Graph of the ZZ-function along the critical line