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Properties

Label 2-200-1.1-c3-0-8
Degree 22
Conductor 200200
Sign 11
Analytic cond. 11.800311.8003
Root an. cond. 3.435163.43516
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  + 9·3-s + 26·7-s + 54·9-s − 59·11-s + 28·13-s + 5·17-s + 109·19-s + 234·21-s − 194·23-s + 243·27-s − 32·29-s + 10·31-s − 531·33-s − 198·37-s + 252·39-s + 117·41-s + 388·43-s − 68·47-s + 333·49-s + 45·51-s − 18·53-s + 981·57-s + 392·59-s − 710·61-s + 1.40e3·63-s − 253·67-s − 1.74e3·69-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.40·7-s + 2·9-s − 1.61·11-s + 0.597·13-s + 0.0713·17-s + 1.31·19-s + 2.43·21-s − 1.75·23-s + 1.73·27-s − 0.204·29-s + 0.0579·31-s − 2.80·33-s − 0.879·37-s + 1.03·39-s + 0.445·41-s + 1.37·43-s − 0.211·47-s + 0.970·49-s + 0.123·51-s − 0.0466·53-s + 2.27·57-s + 0.864·59-s − 1.49·61-s + 2.80·63-s − 0.461·67-s − 3.04·69-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 11.800311.8003
Root analytic conductor: 3.435163.43516
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 200, ( :3/2), 1)(2,\ 200,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.3653711563.365371156
L(12)L(\frac12) ≈\approx 3.3653711563.365371156
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−p2T+p3T2 1 - p^{2} T + p^{3} T^{2}
7 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
11 1+59T+p3T2 1 + 59 T + p^{3} T^{2}
13 1−28T+p3T2 1 - 28 T + p^{3} T^{2}
17 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
19 1−109T+p3T2 1 - 109 T + p^{3} T^{2}
23 1+194T+p3T2 1 + 194 T + p^{3} T^{2}
29 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
31 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
37 1+198T+p3T2 1 + 198 T + p^{3} T^{2}
41 1−117T+p3T2 1 - 117 T + p^{3} T^{2}
43 1−388T+p3T2 1 - 388 T + p^{3} T^{2}
47 1+68T+p3T2 1 + 68 T + p^{3} T^{2}
53 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
59 1−392T+p3T2 1 - 392 T + p^{3} T^{2}
61 1+710T+p3T2 1 + 710 T + p^{3} T^{2}
67 1+253T+p3T2 1 + 253 T + p^{3} T^{2}
71 1+612T+p3T2 1 + 612 T + p^{3} T^{2}
73 1+549T+p3T2 1 + 549 T + p^{3} T^{2}
79 1−414T+p3T2 1 - 414 T + p^{3} T^{2}
83 1+121T+p3T2 1 + 121 T + p^{3} T^{2}
89 1+81T+p3T2 1 + 81 T + p^{3} T^{2}
97 1+1502T+p3T2 1 + 1502 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

−12.13021354758249709631225017720, −10.85022621301508700998467311260, −9.916970747674976432630195441636, −8.774271175234734415392381115984, −7.908171117331579672088646876752, −7.58474050625036628826384153412, −5.47246513530206703464052781374, −4.18254549874557707237059706123, −2.84374232689199147475926810422, −1.70020505708268590747565550696, 1.70020505708268590747565550696, 2.84374232689199147475926810422, 4.18254549874557707237059706123, 5.47246513530206703464052781374, 7.58474050625036628826384153412, 7.908171117331579672088646876752, 8.774271175234734415392381115984, 9.916970747674976432630195441636, 10.85022621301508700998467311260, 12.13021354758249709631225017720

Graph of the ZZ-function along the critical line