↑

Properties

Label 2-40e2-1.1-c3-0-100
Degree 22
Conductor 16001600
Sign −1-1
Analytic cond. 94.403094.4030
Root an. cond. 9.716129.71612
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s + 10·7-s − 2·9-s + 15·11-s − 8·13-s − 21·17-s − 105·19-s + 50·21-s + 10·23-s − 145·27-s + 20·29-s − 230·31-s + 75·33-s + 54·37-s − 40·39-s − 195·41-s − 300·43-s + 480·47-s − 243·49-s − 105·51-s − 322·53-s − 525·57-s − 560·59-s + 730·61-s − 20·63-s + 255·67-s + 50·69-s + ⋯
L(s)  = 1  + 0.962·3-s + 0.539·7-s − 0.0740·9-s + 0.411·11-s − 0.170·13-s − 0.299·17-s − 1.26·19-s + 0.519·21-s + 0.0906·23-s − 1.03·27-s + 0.128·29-s − 1.33·31-s + 0.395·33-s + 0.239·37-s − 0.164·39-s − 0.742·41-s − 1.06·43-s + 1.48·47-s − 0.708·49-s − 0.288·51-s − 0.834·53-s − 1.21·57-s − 1.23·59-s + 1.53·61-s − 0.0399·63-s + 0.464·67-s + 0.0872·69-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 94.403094.4030
Root analytic conductor: 9.716129.71612
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1600, ( :3/2), −1)(2,\ 1600,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
7 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
11 1−15T+p3T2 1 - 15 T + p^{3} T^{2}
13 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
17 1+21T+p3T2 1 + 21 T + p^{3} T^{2}
19 1+105T+p3T2 1 + 105 T + p^{3} T^{2}
23 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
29 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
31 1+230T+p3T2 1 + 230 T + p^{3} T^{2}
37 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
41 1+195T+p3T2 1 + 195 T + p^{3} T^{2}
43 1+300T+p3T2 1 + 300 T + p^{3} T^{2}
47 1−480T+p3T2 1 - 480 T + p^{3} T^{2}
53 1+322T+p3T2 1 + 322 T + p^{3} T^{2}
59 1+560T+p3T2 1 + 560 T + p^{3} T^{2}
61 1−730T+p3T2 1 - 730 T + p^{3} T^{2}
67 1−255T+p3T2 1 - 255 T + p^{3} T^{2}
71 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
73 1−317T+p3T2 1 - 317 T + p^{3} T^{2}
79 1+830T+p3T2 1 + 830 T + p^{3} T^{2}
83 1−75T+p3T2 1 - 75 T + p^{3} T^{2}
89 1+705T+p3T2 1 + 705 T + p^{3} T^{2}
97 1+1434T+p3T2 1 + 1434 T + p^{3} 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

−8.637085686210801865478384767505, −8.047676659695415522388197185631, −7.17784792113914659573016817718, −6.29632280013500323434427179475, −5.27686679882024616118356197121, −4.29233684105911263283829717725, −3.47086758258914579908210703791, −2.41784539479850869364425643068, −1.62326759629559723688036910136, 0, 1.62326759629559723688036910136, 2.41784539479850869364425643068, 3.47086758258914579908210703791, 4.29233684105911263283829717725, 5.27686679882024616118356197121, 6.29632280013500323434427179475, 7.17784792113914659573016817718, 8.047676659695415522388197185631, 8.637085686210801865478384767505

Graph of the ZZ-function along the critical line