↑

Properties

Label 2-40e2-1.1-c1-0-5
Degree 22
Conductor 16001600
Sign 11
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
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-s + 2·7-s − 2·9-s − 3·11-s − 4·13-s + 3·17-s + 5·19-s − 2·21-s + 6·23-s + 5·27-s − 2·31-s + 3·33-s + 2·37-s + 4·39-s − 3·41-s + 4·43-s + 12·47-s − 3·49-s − 3·51-s + 6·53-s − 5·57-s − 2·61-s − 4·63-s + 13·67-s − 6·69-s − 12·71-s − 11·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s − 2/3·9-s − 0.904·11-s − 1.10·13-s + 0.727·17-s + 1.14·19-s − 0.436·21-s + 1.25·23-s + 0.962·27-s − 0.359·31-s + 0.522·33-s + 0.328·37-s + 0.640·39-s − 0.468·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s − 0.420·51-s + 0.824·53-s − 0.662·57-s − 0.256·61-s − 0.503·63-s + 1.58·67-s − 0.722·69-s − 1.42·71-s − 1.28·73-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 11
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 1)(2,\ 1600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2629926601.262992660
L(12)L(\frac12) ≈\approx 1.2629926601.262992660
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 1
good3 1+T+pT2 1 + T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−13T+pT2 1 - 13 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1−15T+pT2 1 - 15 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−9.429239826768018761346169045309, −8.592914531793421630007930470901, −7.63971798893296337235505790724, −7.23606762344534562872845833638, −5.91956414482903908532570774168, −5.22748830252304089500003968022, −4.78507259529948280984941273759, −3.27747260558597777923671030481, −2.37228414250853175167766648204, −0.806837847292090646690279005746, 0.806837847292090646690279005746, 2.37228414250853175167766648204, 3.27747260558597777923671030481, 4.78507259529948280984941273759, 5.22748830252304089500003968022, 5.91956414482903908532570774168, 7.23606762344534562872845833638, 7.63971798893296337235505790724, 8.592914531793421630007930470901, 9.429239826768018761346169045309

Graph of the ZZ-function along the critical line