↑

Properties

Label 2-3840-1.1-c1-0-20
Degree 22
Conductor 38403840
Sign 11
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s + 6·13-s − 15-s − 2·19-s + 6·23-s + 25-s + 27-s + 6·29-s − 6·37-s + 6·39-s − 6·41-s − 8·43-s − 45-s − 6·47-s − 7·49-s + 6·53-s − 2·57-s + 12·59-s + 12·61-s − 6·65-s − 4·67-s + 6·69-s + 12·71-s − 2·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.66·13-s − 0.258·15-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.986·37-s + 0.960·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.875·47-s − 49-s + 0.824·53-s − 0.264·57-s + 1.56·59-s + 1.53·61-s − 0.744·65-s − 0.488·67-s + 0.722·69-s + 1.42·71-s − 0.234·73-s + 0.115·75-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3840, ( :1/2), 1)(2,\ 3840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4052873332.405287333
L(12)L(\frac12) ≈\approx 2.4052873332.405287333
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
3 1−T 1 - T
5 1+T 1 + T
good7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+pT2 1 + 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+6T+pT2 1 + 6 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−6T+pT2 1 - 6 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

−8.491558126432897481987602583832, −8.025558724206857712517754972973, −6.82033201421353101559868448454, −6.61458075237655978501027495472, −5.40631980079267041264148366817, −4.63550796724991546581859348625, −3.63216606310010362585347506817, −3.22642268183474511512896931271, −1.97776141782979305959520401409, −0.909537853146744575546185673329, 0.909537853146744575546185673329, 1.97776141782979305959520401409, 3.22642268183474511512896931271, 3.63216606310010362585347506817, 4.63550796724991546581859348625, 5.40631980079267041264148366817, 6.61458075237655978501027495472, 6.82033201421353101559868448454, 8.025558724206857712517754972973, 8.491558126432897481987602583832

Graph of the ZZ-function along the critical line