↑

Properties

Label 2-40e2-1.1-c3-0-34
Degree 22
Conductor 16001600
Sign 11
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 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 7·3-s + 6·7-s + 22·9-s + 43·11-s + 28·13-s + 91·17-s + 35·19-s − 42·21-s + 162·23-s + 35·27-s − 160·29-s + 42·31-s − 301·33-s + 314·37-s − 196·39-s − 203·41-s − 92·43-s + 196·47-s − 307·49-s − 637·51-s − 82·53-s − 245·57-s + 280·59-s + 518·61-s + 132·63-s − 141·67-s − 1.13e3·69-s + ⋯
L(s)  = 1  − 1.34·3-s + 0.323·7-s + 0.814·9-s + 1.17·11-s + 0.597·13-s + 1.29·17-s + 0.422·19-s − 0.436·21-s + 1.46·23-s + 0.249·27-s − 1.02·29-s + 0.243·31-s − 1.58·33-s + 1.39·37-s − 0.804·39-s − 0.773·41-s − 0.326·43-s + 0.608·47-s − 0.895·49-s − 1.74·51-s − 0.212·53-s − 0.569·57-s + 0.617·59-s + 1.08·61-s + 0.263·63-s − 0.257·67-s − 1.97·69-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: 11
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: 00
Selberg data: (2, 1600, ( :3/2), 1)(2,\ 1600,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.7293670921.729367092
L(12)L(\frac12) ≈\approx 1.7293670921.729367092
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+7T+p3T2 1 + 7 T + p^{3} T^{2}
7 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1−43T+p3T2 1 - 43 T + p^{3} T^{2}
13 1−28T+p3T2 1 - 28 T + p^{3} T^{2}
17 1−91T+p3T2 1 - 91 T + p^{3} T^{2}
19 1−35T+p3T2 1 - 35 T + p^{3} T^{2}
23 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
29 1+160T+p3T2 1 + 160 T + p^{3} T^{2}
31 1−42T+p3T2 1 - 42 T + p^{3} T^{2}
37 1−314T+p3T2 1 - 314 T + p^{3} T^{2}
41 1+203T+p3T2 1 + 203 T + p^{3} T^{2}
43 1+92T+p3T2 1 + 92 T + p^{3} T^{2}
47 1−196T+p3T2 1 - 196 T + p^{3} T^{2}
53 1+82T+p3T2 1 + 82 T + p^{3} T^{2}
59 1−280T+p3T2 1 - 280 T + p^{3} T^{2}
61 1−518T+p3T2 1 - 518 T + p^{3} T^{2}
67 1+141T+p3T2 1 + 141 T + p^{3} T^{2}
71 1−412T+p3T2 1 - 412 T + p^{3} T^{2}
73 1+763T+p3T2 1 + 763 T + p^{3} T^{2}
79 1−510T+p3T2 1 - 510 T + p^{3} T^{2}
83 1+777T+p3T2 1 + 777 T + p^{3} T^{2}
89 1+945T+p3T2 1 + 945 T + p^{3} T^{2}
97 1−1246T+p3T2 1 - 1246 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

−9.162807130780010267449038509392, −8.220536188033643559884166462784, −7.22399100219122118214443336759, −6.52221224936827348001238382577, −5.72654269895957227979150128556, −5.13122971338231583607225141514, −4.13721208209058985945257453930, −3.14836832984906747400199666811, −1.44017897451635070895122931497, −0.77454565099657160217588920506, 0.77454565099657160217588920506, 1.44017897451635070895122931497, 3.14836832984906747400199666811, 4.13721208209058985945257453930, 5.13122971338231583607225141514, 5.72654269895957227979150128556, 6.52221224936827348001238382577, 7.22399100219122118214443336759, 8.220536188033643559884166462784, 9.162807130780010267449038509392

Graph of the ZZ-function along the critical line