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Properties

Label 2-560-1.1-c5-0-6
Degree 22
Conductor 560560
Sign 11
Analytic cond. 89.814989.8149
Root an. cond. 9.477079.47707
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 25·5-s − 49·7-s − 234·9-s − 405·11-s − 391·13-s − 75·15-s + 999·17-s − 2.34e3·19-s − 147·21-s − 2.43e3·23-s + 625·25-s − 1.43e3·27-s + 8.25e3·29-s − 4.01e3·31-s − 1.21e3·33-s + 1.22e3·35-s − 7.04e3·37-s − 1.17e3·39-s + 3.33e3·41-s + 2.35e4·43-s + 5.85e3·45-s − 1.03e4·47-s + 2.40e3·49-s + 2.99e3·51-s + 3.08e3·53-s + 1.01e4·55-s + ⋯
L(s)  = 1  + 0.192·3-s − 0.447·5-s − 0.377·7-s − 0.962·9-s − 1.00·11-s − 0.641·13-s − 0.0860·15-s + 0.838·17-s − 1.48·19-s − 0.0727·21-s − 0.957·23-s + 1/5·25-s − 0.377·27-s + 1.82·29-s − 0.750·31-s − 0.194·33-s + 0.169·35-s − 0.845·37-s − 0.123·39-s + 0.309·41-s + 1.93·43-s + 0.430·45-s − 0.681·47-s + 1/7·49-s + 0.161·51-s + 0.150·53-s + 0.451·55-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: 11
Analytic conductor: 89.814989.8149
Root analytic conductor: 9.477079.47707
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 560, ( :5/2), 1)(2,\ 560,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.86729050010.8672905001
L(12)L(\frac12) ≈\approx 0.86729050010.8672905001
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+p2T 1 + p^{2} T
7 1+p2T 1 + p^{2} T
good3 1−pT+p5T2 1 - p T + p^{5} T^{2}
11 1+405T+p5T2 1 + 405 T + p^{5} T^{2}
13 1+391T+p5T2 1 + 391 T + p^{5} T^{2}
17 1−999T+p5T2 1 - 999 T + p^{5} T^{2}
19 1+2342T+p5T2 1 + 2342 T + p^{5} T^{2}
23 1+2430T+p5T2 1 + 2430 T + p^{5} T^{2}
29 1−8259T+p5T2 1 - 8259 T + p^{5} T^{2}
31 1+4016T+p5T2 1 + 4016 T + p^{5} T^{2}
37 1+7042T+p5T2 1 + 7042 T + p^{5} T^{2}
41 1−3336T+p5T2 1 - 3336 T + p^{5} T^{2}
43 1−23518T+p5T2 1 - 23518 T + p^{5} T^{2}
47 1+10317T+p5T2 1 + 10317 T + p^{5} T^{2}
53 1−3084T+p5T2 1 - 3084 T + p^{5} T^{2}
59 1−18816T+p5T2 1 - 18816 T + p^{5} T^{2}
61 1−21668T+p5T2 1 - 21668 T + p^{5} T^{2}
67 1+52124T+p5T2 1 + 52124 T + p^{5} T^{2}
71 1−28560T+p5T2 1 - 28560 T + p^{5} T^{2}
73 1+70342T+p5T2 1 + 70342 T + p^{5} T^{2}
79 1+58823T+p5T2 1 + 58823 T + p^{5} T^{2}
83 1+756T+p5T2 1 + 756 T + p^{5} T^{2}
89 1−135384T+p5T2 1 - 135384 T + p^{5} T^{2}
97 1−110435T+p5T2 1 - 110435 T + p^{5} T^{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

−10.15192115795869857838353514070, −8.948314167369957459276500043997, −8.208200243308395796302434846113, −7.47129907455081247803957419187, −6.28922551278524797523438370732, −5.39751006135097991580984639624, −4.28094725904091955264225850362, −3.09567909604928904139392292701, −2.26475194427441717514435378588, −0.42496416859401474821204349671, 0.42496416859401474821204349671, 2.26475194427441717514435378588, 3.09567909604928904139392292701, 4.28094725904091955264225850362, 5.39751006135097991580984639624, 6.28922551278524797523438370732, 7.47129907455081247803957419187, 8.208200243308395796302434846113, 8.948314167369957459276500043997, 10.15192115795869857838353514070

Graph of the ZZ-function along the critical line