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Properties

Label 2-200-1.1-c1-0-1
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  + 4·7-s − 3·9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s − 4·23-s − 2·29-s − 8·31-s − 6·37-s − 6·41-s + 8·43-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s − 2·61-s − 12·63-s − 8·67-s + 6·73-s + 16·77-s + 9·81-s + 16·83-s − 6·89-s + 8·91-s + 14·97-s − 12·99-s + ⋯
L(s)  = 1  + 1.51·7-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + 1.75·83-s − 0.635·89-s + 0.838·91-s + 1.42·97-s − 1.20·99-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 1.3276988781.327698878
L(12)L(\frac12) ≈\approx 1.3276988781.327698878
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+pT2 1 + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−12.16106454520843121806724596230, −11.45666669128344280655084086532, −10.83635878381861701657899519690, −9.267253735733507250496452791400, −8.516348691113562212594470236153, −7.50932176073420730404294525891, −6.11165748014924370206663675456, −5.01978566915858300867155671122, −3.66795927025167344126272800405, −1.73185811467864323225134521099, 1.73185811467864323225134521099, 3.66795927025167344126272800405, 5.01978566915858300867155671122, 6.11165748014924370206663675456, 7.50932176073420730404294525891, 8.516348691113562212594470236153, 9.267253735733507250496452791400, 10.83635878381861701657899519690, 11.45666669128344280655084086532, 12.16106454520843121806724596230

Graph of the ZZ-function along the critical line