↑

Properties

Label 2-52e2-1.1-c1-0-38
Degree 22
Conductor 27042704
Sign 11
Analytic cond. 21.591521.5915
Root an. cond. 4.646674.64667
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  + 3·3-s + 5-s + 7-s + 6·9-s − 2·11-s + 3·15-s − 3·17-s + 6·19-s + 3·21-s + 4·23-s − 4·25-s + 9·27-s + 2·29-s + 4·31-s − 6·33-s + 35-s − 3·37-s + 5·43-s + 6·45-s + 13·47-s − 6·49-s − 9·51-s + 12·53-s − 2·55-s + 18·57-s − 10·59-s − 8·61-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 0.603·11-s + 0.774·15-s − 0.727·17-s + 1.37·19-s + 0.654·21-s + 0.834·23-s − 4/5·25-s + 1.73·27-s + 0.371·29-s + 0.718·31-s − 1.04·33-s + 0.169·35-s − 0.493·37-s + 0.762·43-s + 0.894·45-s + 1.89·47-s − 6/7·49-s − 1.26·51-s + 1.64·53-s − 0.269·55-s + 2.38·57-s − 1.30·59-s − 1.02·61-s + ⋯

Functional equation

Λ(s)=(2704s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2704s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27042704    =    24⋅1322^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 21.591521.5915
Root analytic conductor: 4.646674.64667
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2704, ( :1/2), 1)(2,\ 2704,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.0028435164.002843516
L(12)L(\frac12) ≈\approx 4.0028435164.002843516
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
13 1 1
good3 1−pT+pT2 1 - p T + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−6T+pT2 1 - 6 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 1−4T+pT2 1 - 4 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1−13T+pT2 1 - 13 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+pT2 1 + 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

−8.968437800976028493256072760850, −8.051660281056901581015263821725, −7.58433170795290376453101971689, −6.83942202332506672968213427091, −5.67644357149883273045642983440, −4.77110978952904042995749180644, −3.89239162271605294968540111677, −2.91875359123950367684627079786, −2.35658502852942708851276680816, −1.29336546110615612462449119400, 1.29336546110615612462449119400, 2.35658502852942708851276680816, 2.91875359123950367684627079786, 3.89239162271605294968540111677, 4.77110978952904042995749180644, 5.67644357149883273045642983440, 6.83942202332506672968213427091, 7.58433170795290376453101971689, 8.051660281056901581015263821725, 8.968437800976028493256072760850

Graph of the ZZ-function along the critical line