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Properties

Label 2-20e2-1.1-c3-0-3
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  − 2·3-s − 26·7-s − 23·9-s + 28·11-s + 12·13-s − 64·17-s + 60·19-s + 52·21-s + 58·23-s + 100·27-s + 90·29-s + 128·31-s − 56·33-s + 236·37-s − 24·39-s + 242·41-s − 362·43-s − 226·47-s + 333·49-s + 128·51-s − 108·53-s − 120·57-s + 20·59-s + 542·61-s + 598·63-s + 434·67-s − 116·69-s + ⋯
L(s)  = 1  − 0.384·3-s − 1.40·7-s − 0.851·9-s + 0.767·11-s + 0.256·13-s − 0.913·17-s + 0.724·19-s + 0.540·21-s + 0.525·23-s + 0.712·27-s + 0.576·29-s + 0.741·31-s − 0.295·33-s + 1.04·37-s − 0.0985·39-s + 0.921·41-s − 1.28·43-s − 0.701·47-s + 0.970·49-s + 0.351·51-s − 0.279·53-s − 0.278·57-s + 0.0441·59-s + 1.13·61-s + 1.19·63-s + 0.791·67-s − 0.202·69-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 1.1526356861.152635686
L(12)L(\frac12) ≈\approx 1.1526356861.152635686
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+2T+p3T2 1 + 2 T + p^{3} T^{2}
7 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
11 1−28T+p3T2 1 - 28 T + p^{3} T^{2}
13 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
17 1+64T+p3T2 1 + 64 T + p^{3} T^{2}
19 1−60T+p3T2 1 - 60 T + p^{3} T^{2}
23 1−58T+p3T2 1 - 58 T + p^{3} T^{2}
29 1−90T+p3T2 1 - 90 T + p^{3} T^{2}
31 1−128T+p3T2 1 - 128 T + p^{3} T^{2}
37 1−236T+p3T2 1 - 236 T + p^{3} T^{2}
41 1−242T+p3T2 1 - 242 T + p^{3} T^{2}
43 1+362T+p3T2 1 + 362 T + p^{3} T^{2}
47 1+226T+p3T2 1 + 226 T + p^{3} T^{2}
53 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
59 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
61 1−542T+p3T2 1 - 542 T + p^{3} T^{2}
67 1−434T+p3T2 1 - 434 T + p^{3} T^{2}
71 1−1128T+p3T2 1 - 1128 T + p^{3} T^{2}
73 1−632T+p3T2 1 - 632 T + p^{3} T^{2}
79 1−720T+p3T2 1 - 720 T + p^{3} T^{2}
83 1−478T+p3T2 1 - 478 T + p^{3} T^{2}
89 1+490T+p3T2 1 + 490 T + p^{3} T^{2}
97 1−1456T+p3T2 1 - 1456 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

−10.99384244587925400599830176806, −9.816565172266707057540172373269, −9.185645826282686450801510495935, −8.181782643830396162926701350027, −6.66858873074558629658346091442, −6.34713158720880303990299397279, −5.10755513074204629174673699580, −3.72181200667750156606665190485, −2.69188977380543605078092619993, −0.69824173013547367662006809369, 0.69824173013547367662006809369, 2.69188977380543605078092619993, 3.72181200667750156606665190485, 5.10755513074204629174673699580, 6.34713158720880303990299397279, 6.66858873074558629658346091442, 8.181782643830396162926701350027, 9.185645826282686450801510495935, 9.816565172266707057540172373269, 10.99384244587925400599830176806

Graph of the ZZ-function along the critical line