↑

Properties

Label 2-880-1.1-c1-0-6
Degree 22
Conductor 880880
Sign 11
Analytic cond. 7.026837.02683
Root an. cond. 2.650812.65081
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3·9-s + 11-s + 2·13-s + 6·17-s + 4·19-s − 4·23-s + 25-s + 6·29-s + 8·31-s − 2·37-s + 2·41-s − 4·43-s − 3·45-s + 12·47-s − 7·49-s − 2·53-s + 55-s − 4·59-s − 10·61-s + 2·65-s + 16·67-s − 8·71-s + 14·73-s − 8·79-s + 9·81-s + 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 9-s + 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.75·47-s − 49-s − 0.274·53-s + 0.134·55-s − 0.520·59-s − 1.28·61-s + 0.248·65-s + 1.95·67-s − 0.949·71-s + 1.63·73-s − 0.900·79-s + 81-s + 0.439·83-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 7.026837.02683
Root analytic conductor: 2.650812.65081
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 880, ( :1/2), 1)(2,\ 880,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7212734481.721273448
L(12)L(\frac12) ≈\approx 1.7212734481.721273448
L(32)L(\frac{3}{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−T 1 - T
11 1−T 1 - T
good3 1+pT2 1 + p T^{2}
7 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p 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

−10.07588915028781859505118646541, −9.376036497146997809466572379928, −8.402071812243779349662792300606, −7.78041223670494609407513138154, −6.49908708730306794178412805091, −5.83555924118533390298503875284, −4.97617253141282559040022801022, −3.60799086718076388676488423337, −2.69303075235727048947048138558, −1.13555459814448309624758433433, 1.13555459814448309624758433433, 2.69303075235727048947048138558, 3.60799086718076388676488423337, 4.97617253141282559040022801022, 5.83555924118533390298503875284, 6.49908708730306794178412805091, 7.78041223670494609407513138154, 8.402071812243779349662792300606, 9.376036497146997809466572379928, 10.07588915028781859505118646541

Graph of the ZZ-function along the critical line