↑

Properties

Label 2-560-1.1-c5-0-13
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 11·3-s − 25·5-s − 49·7-s − 122·9-s + 267·11-s − 1.08e3·13-s − 275·15-s − 513·17-s + 802·19-s − 539·21-s + 1.29e3·23-s + 625·25-s − 4.01e3·27-s + 1.77e3·29-s + 2.58e3·31-s + 2.93e3·33-s + 1.22e3·35-s + 1.38e4·37-s − 1.19e4·39-s − 1.19e4·41-s + 598·43-s + 3.05e3·45-s + 1.70e4·47-s + 2.40e3·49-s − 5.64e3·51-s + 2.78e4·53-s − 6.67e3·55-s + ⋯
L(s)  = 1  + 0.705·3-s − 0.447·5-s − 0.377·7-s − 0.502·9-s + 0.665·11-s − 1.78·13-s − 0.315·15-s − 0.430·17-s + 0.509·19-s − 0.266·21-s + 0.508·23-s + 1/5·25-s − 1.05·27-s + 0.392·29-s + 0.482·31-s + 0.469·33-s + 0.169·35-s + 1.66·37-s − 1.25·39-s − 1.10·41-s + 0.0493·43-s + 0.224·45-s + 1.12·47-s + 1/7·49-s − 0.303·51-s + 1.36·53-s − 0.297·55-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 1.8406076161.840607616
L(12)L(\frac12) ≈\approx 1.8406076161.840607616
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 1−11T+p5T2 1 - 11 T + p^{5} T^{2}
11 1−267T+p5T2 1 - 267 T + p^{5} T^{2}
13 1+1087T+p5T2 1 + 1087 T + p^{5} T^{2}
17 1+513T+p5T2 1 + 513 T + p^{5} T^{2}
19 1−802T+p5T2 1 - 802 T + p^{5} T^{2}
23 1−1290T+p5T2 1 - 1290 T + p^{5} T^{2}
29 1−1779T+p5T2 1 - 1779 T + p^{5} T^{2}
31 1−2584T+p5T2 1 - 2584 T + p^{5} T^{2}
37 1−13862T+p5T2 1 - 13862 T + p^{5} T^{2}
41 1+11904T+p5T2 1 + 11904 T + p^{5} T^{2}
43 1−598T+p5T2 1 - 598 T + p^{5} T^{2}
47 1−17019T+p5T2 1 - 17019 T + p^{5} T^{2}
53 1−27852T+p5T2 1 - 27852 T + p^{5} T^{2}
59 1+30912T+p5T2 1 + 30912 T + p^{5} T^{2}
61 1+1780T+p5T2 1 + 1780 T + p^{5} T^{2}
67 1+25052T+p5T2 1 + 25052 T + p^{5} T^{2}
71 1−51984T+p5T2 1 - 51984 T + p^{5} T^{2}
73 1−47690T+p5T2 1 - 47690 T + p^{5} T^{2}
79 1−102121T+p5T2 1 - 102121 T + p^{5} T^{2}
83 1−83676T+p5T2 1 - 83676 T + p^{5} T^{2}
89 1+32400T+p5T2 1 + 32400 T + p^{5} T^{2}
97 1+148645T+p5T2 1 + 148645 T + p^{5} T^{2}
show more
show less
   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

−9.685217489136929876532401482308, −9.203772050057122624776058999023, −8.197803174520189394061605679034, −7.40376269087334489308447048222, −6.52798668033269083030857023849, −5.24683370626312080338183158694, −4.20857848538291660616281787201, −3.08719208046657794689392510134, −2.30620156561801784641769181084, −0.62044504649449934031384600739, 0.62044504649449934031384600739, 2.30620156561801784641769181084, 3.08719208046657794689392510134, 4.20857848538291660616281787201, 5.24683370626312080338183158694, 6.52798668033269083030857023849, 7.40376269087334489308447048222, 8.197803174520189394061605679034, 9.203772050057122624776058999023, 9.685217489136929876532401482308

Graph of the ZZ-function along the critical line