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Properties

Label 2-20e2-1.1-c3-0-11
Degree 22
Conductor 400400
Sign 11
Analytic cond. 23.600723.6007
Root an. cond. 4.858064.85806
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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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)=(400s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 23.600723.6007
Root analytic conductor: 4.858064.85806
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 400, ( :3/2), 1)(2,\ 400,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.4257626203.425762620
L(12)L(\frac12) ≈\approx 3.4257626203.425762620
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}
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   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

−10.92081425850380038148080963003, −9.580792465687967311626525223190, −8.937294796964111206152927306488, −8.337588735072994521484418993779, −7.22555639822996190597255508028, −6.29697977567539301089455223173, −4.70001234491495502018202021736, −3.68542305177136683026914559285, −2.61052767868810440919122369239, −1.29658064996042049535564465120, 1.29658064996042049535564465120, 2.61052767868810440919122369239, 3.68542305177136683026914559285, 4.70001234491495502018202021736, 6.29697977567539301089455223173, 7.22555639822996190597255508028, 8.337588735072994521484418993779, 8.937294796964111206152927306488, 9.580792465687967311626525223190, 10.92081425850380038148080963003

Graph of the ZZ-function along the critical line