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Properties

Label 2-200-1.1-c1-0-2
Degree 22
Conductor 200200
Sign 11
Analytic cond. 1.597001.59700
Root an. cond. 1.263721.26372
Motivic weight 11
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 + 2·7-s + 9-s − 4·11-s + 4·13-s − 4·19-s + 4·21-s − 2·23-s − 4·27-s + 2·29-s − 8·33-s + 4·37-s + 8·39-s + 2·41-s − 6·43-s − 6·47-s − 3·49-s − 4·53-s − 8·57-s − 12·59-s − 10·61-s + 2·63-s + 14·67-s − 4·69-s + 8·71-s + 8·73-s − 8·77-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.917·19-s + 0.872·21-s − 0.417·23-s − 0.769·27-s + 0.371·29-s − 1.39·33-s + 0.657·37-s + 1.28·39-s + 0.312·41-s − 0.914·43-s − 0.875·47-s − 3/7·49-s − 0.549·53-s − 1.05·57-s − 1.56·59-s − 1.28·61-s + 0.251·63-s + 1.71·67-s − 0.481·69-s + 0.949·71-s + 0.936·73-s − 0.911·77-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 1.597001.59700
Root analytic conductor: 1.263721.26372
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 200, ( :1/2), 1)(2,\ 200,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6601084181.660108418
L(12)L(\frac12) ≈\approx 1.6601084181.660108418
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 1−2T+pT2 1 - 2 T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−16T+pT2 1 - 16 T + p 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

−12.73146010216774514797104134533, −11.33359133996889118176969030818, −10.53622600687691340454258542756, −9.301178546757553526958267434221, −8.212252834472441096002849564389, −7.930378831312213533244284917039, −6.25352345843664090435411695863, −4.82604799709023040346819665152, −3.41809103272636300318248682927, −2.08793608524985768454862433122, 2.08793608524985768454862433122, 3.41809103272636300318248682927, 4.82604799709023040346819665152, 6.25352345843664090435411695863, 7.930378831312213533244284917039, 8.212252834472441096002849564389, 9.301178546757553526958267434221, 10.53622600687691340454258542756, 11.33359133996889118176969030818, 12.73146010216774514797104134533

Graph of the ZZ-function along the critical line