↑

Properties

Label 2-9280-1.1-c1-0-100
Degree 22
Conductor 92809280
Sign 11
Analytic cond. 74.101174.1011
Root an. cond. 8.608208.60820
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 + 2·7-s − 3·9-s + 2·11-s + 6·13-s + 2·17-s − 2·19-s + 6·23-s + 25-s + 29-s + 6·31-s + 2·35-s + 2·37-s + 10·41-s − 8·43-s − 3·45-s + 4·47-s − 3·49-s − 10·53-s + 2·55-s + 8·59-s − 10·61-s − 6·63-s + 6·65-s + 2·67-s − 4·71-s + 6·73-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s − 9-s + 0.603·11-s + 1.66·13-s + 0.485·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.185·29-s + 1.07·31-s + 0.338·35-s + 0.328·37-s + 1.56·41-s − 1.21·43-s − 0.447·45-s + 0.583·47-s − 3/7·49-s − 1.37·53-s + 0.269·55-s + 1.04·59-s − 1.28·61-s − 0.755·63-s + 0.744·65-s + 0.244·67-s − 0.474·71-s + 0.702·73-s + ⋯

Functional equation

Λ(s)=(9280s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9280s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92809280    =    26⋅5⋅292^{6} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 74.101174.1011
Root analytic conductor: 8.608208.60820
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9280, ( :1/2), 1)(2,\ 9280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9580549732.958054973
L(12)L(\frac12) ≈\approx 2.9580549732.958054973
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
29 1−T 1 - T
good3 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−10T+pT2 1 - 10 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+10T+pT2 1 + 10 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−8.052840808513428859574580540510, −6.85129093115692814292716553751, −6.28660003535746607565691278176, −5.74840797572829895528228917775, −5.00639051128266281017919416554, −4.24721196488992332309824332666, −3.36994472218581743273921077000, −2.68401564538192621761486313487, −1.58564699459202598374088008614, −0.904981875417245622946714514503, 0.904981875417245622946714514503, 1.58564699459202598374088008614, 2.68401564538192621761486313487, 3.36994472218581743273921077000, 4.24721196488992332309824332666, 5.00639051128266281017919416554, 5.74840797572829895528228917775, 6.28660003535746607565691278176, 6.85129093115692814292716553751, 8.052840808513428859574580540510

Graph of the ZZ-function along the critical line