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Properties

Label 2-9200-1.1-c1-0-16
Degree 22
Conductor 92009200
Sign 11
Analytic cond. 73.462373.4623
Root an. cond. 8.571018.57101
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  − 3·3-s − 2·7-s + 6·9-s − 13-s + 6·21-s + 23-s − 9·27-s − 3·29-s − 3·31-s + 8·37-s + 3·39-s + 3·41-s − 2·43-s − 11·47-s − 3·49-s + 14·53-s + 8·59-s − 4·61-s − 12·63-s − 4·67-s − 3·69-s − 7·71-s + 9·73-s + 9·81-s + 4·83-s + 9·87-s − 2·89-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.755·7-s + 2·9-s − 0.277·13-s + 1.30·21-s + 0.208·23-s − 1.73·27-s − 0.557·29-s − 0.538·31-s + 1.31·37-s + 0.480·39-s + 0.468·41-s − 0.304·43-s − 1.60·47-s − 3/7·49-s + 1.92·53-s + 1.04·59-s − 0.512·61-s − 1.51·63-s − 0.488·67-s − 0.361·69-s − 0.830·71-s + 1.05·73-s + 81-s + 0.439·83-s + 0.964·87-s − 0.211·89-s + ⋯

Functional equation

Λ(s)=(9200s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9200s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92009200    =    24⋅52⋅232^{4} \cdot 5^{2} \cdot 23
Sign: 11
Analytic conductor: 73.462373.4623
Root analytic conductor: 8.571018.57101
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9200, ( :1/2), 1)(2,\ 9200,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.57709591800.5770959180
L(12)L(\frac12) ≈\approx 0.57709591800.5770959180
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
23 1−T 1 - T
good3 1+pT+pT2 1 + p T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+11T+pT2 1 + 11 T + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1−9T+pT2 1 - 9 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1+18T+pT2 1 + 18 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

−7.43375077985563887978169616802, −6.84375350435015559761943384325, −6.31797398354246854766794516230, −5.67904551637868855962856681300, −5.14555431991860518244362280094, −4.37586317640795451613096196834, −3.65179626455037376248954783203, −2.58623822926810723667775573674, −1.42273224465244581820044878115, −0.42234562898825569737865036719, 0.42234562898825569737865036719, 1.42273224465244581820044878115, 2.58623822926810723667775573674, 3.65179626455037376248954783203, 4.37586317640795451613096196834, 5.14555431991860518244362280094, 5.67904551637868855962856681300, 6.31797398354246854766794516230, 6.84375350435015559761943384325, 7.43375077985563887978169616802

Graph of the ZZ-function along the critical line