↑

Properties

Label 2-320-1.1-c1-0-7
Degree 22
Conductor 320320
Sign −1-1
Analytic cond. 2.555212.55521
Root an. cond. 1.598501.59850
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 3·9-s − 4·11-s + 2·13-s + 2·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s − 8·31-s + 4·35-s − 6·37-s − 6·41-s + 8·43-s + 3·45-s + 4·47-s + 9·49-s − 6·53-s + 4·55-s + 4·59-s + 2·61-s + 12·63-s − 2·65-s − 8·67-s − 6·73-s + 16·77-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s + 1.21·43-s + 0.447·45-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.248·65-s − 0.977·67-s − 0.702·73-s + 1.82·77-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: −1-1
Analytic conductor: 2.555212.55521
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 320, ( :1/2), −1)(2,\ 320,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
good3 1+pT2 1 + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−2T+pT2 1 - 2 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 1−2T+pT2 1 - 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−11.01108015186625578670651780817, −10.39655063289957855259191225662, −9.206637608661514184739644068270, −8.421702762202697929017375581976, −7.28963827504896405934712065825, −6.22671236180701133336548488807, −5.28966562607622137666656232279, −3.64413717916439174649981742998, −2.75667504176180688667939525456, 0, 2.75667504176180688667939525456, 3.64413717916439174649981742998, 5.28966562607622137666656232279, 6.22671236180701133336548488807, 7.28963827504896405934712065825, 8.421702762202697929017375581976, 9.206637608661514184739644068270, 10.39655063289957855259191225662, 11.01108015186625578670651780817

Graph of the ZZ-function along the critical line