↑

Properties

Label 2-3840-1.1-c1-0-3
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 − 2·7-s + 9-s − 2·11-s + 2·13-s + 15-s − 8·17-s + 8·19-s + 2·21-s + 8·23-s + 25-s − 27-s − 10·29-s − 8·31-s + 2·33-s + 2·35-s − 2·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s − 4·47-s − 3·49-s + 8·51-s + 2·53-s + 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 1.94·17-s + 1.83·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.348·33-s + 0.338·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s + 1.12·51-s + 0.274·53-s + 0.269·55-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 0.86287272210.8628727221
L(12)L(\frac12) ≈\approx 0.86287272210.8628727221
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+2T+pT2 1 + 2 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+8T+pT2 1 + 8 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+10T+pT2 1 + 10 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+6T+pT2 1 + 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−8T+pT2 1 - 8 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

−8.638551538442009893577523164070, −7.42453918815745026023592410788, −7.14082162084889437091737778969, −6.31312099042302818818590735279, −5.41051129030844572729002610372, −4.89112307430868275798271944931, −3.75282749942956815280684885618, −3.17592623893454142168104729943, −1.92860669592669570819603903176, −0.53824953394909956001202651788, 0.53824953394909956001202651788, 1.92860669592669570819603903176, 3.17592623893454142168104729943, 3.75282749942956815280684885618, 4.89112307430868275798271944931, 5.41051129030844572729002610372, 6.31312099042302818818590735279, 7.14082162084889437091737778969, 7.42453918815745026023592410788, 8.638551538442009893577523164070

Graph of the ZZ-function along the critical line