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Properties

Label 2-9200-1.1-c1-0-38
Degree 22
Conductor 92009200
Sign 11
Analytic cond. 73.462373.4623
Root an. cond. 8.571018.57101
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.40·3-s + 1.57·7-s − 1.03·9-s − 4.35·11-s − 0.964·13-s + 0.300·17-s + 8.62·19-s − 2.20·21-s − 23-s + 5.65·27-s + 4.76·29-s + 5.59·31-s + 6.11·33-s + 4.38·37-s + 1.35·39-s − 6.62·41-s − 1.72·43-s − 0.687·47-s − 4.53·49-s − 0.421·51-s − 8.05·53-s − 12.1·57-s − 5.74·59-s − 13.6·61-s − 1.61·63-s + 6.49·67-s + 1.40·69-s + ⋯
L(s)  = 1  − 0.810·3-s + 0.593·7-s − 0.343·9-s − 1.31·11-s − 0.267·13-s + 0.0728·17-s + 1.97·19-s − 0.481·21-s − 0.208·23-s + 1.08·27-s + 0.885·29-s + 1.00·31-s + 1.06·33-s + 0.720·37-s + 0.216·39-s − 1.03·41-s − 0.263·43-s − 0.100·47-s − 0.647·49-s − 0.0589·51-s − 1.10·53-s − 1.60·57-s − 0.748·59-s − 1.74·61-s − 0.204·63-s + 0.792·67-s + 0.168·69-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: no
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 1.1793224611.179322461
L(12)L(\frac12) ≈\approx 1.1793224611.179322461
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+1.40T+3T2 1 + 1.40T + 3T^{2}
7 1−1.57T+7T2 1 - 1.57T + 7T^{2}
11 1+4.35T+11T2 1 + 4.35T + 11T^{2}
13 1+0.964T+13T2 1 + 0.964T + 13T^{2}
17 1−0.300T+17T2 1 - 0.300T + 17T^{2}
19 1−8.62T+19T2 1 - 8.62T + 19T^{2}
29 1−4.76T+29T2 1 - 4.76T + 29T^{2}
31 1−5.59T+31T2 1 - 5.59T + 31T^{2}
37 1−4.38T+37T2 1 - 4.38T + 37T^{2}
41 1+6.62T+41T2 1 + 6.62T + 41T^{2}
43 1+1.72T+43T2 1 + 1.72T + 43T^{2}
47 1+0.687T+47T2 1 + 0.687T + 47T^{2}
53 1+8.05T+53T2 1 + 8.05T + 53T^{2}
59 1+5.74T+59T2 1 + 5.74T + 59T^{2}
61 1+13.6T+61T2 1 + 13.6T + 61T^{2}
67 1−6.49T+67T2 1 - 6.49T + 67T^{2}
71 1+9.89T+71T2 1 + 9.89T + 71T^{2}
73 1+6.35T+73T2 1 + 6.35T + 73T^{2}
79 1−6.95T+79T2 1 - 6.95T + 79T^{2}
83 1+0.185T+83T2 1 + 0.185T + 83T^{2}
89 1+1.64T+89T2 1 + 1.64T + 89T^{2}
97 1−9.21T+97T2 1 - 9.21T + 97T^{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

−7.87561251044937135804478764106, −7.04616215580093283140604949226, −6.19296160655540979528124408319, −5.62923163077647156462500404122, −4.85704086691754657723695208883, −4.72609009512419540977989601285, −3.19896036037118686763361609116, −2.80454738554761168256656770725, −1.57722757377659723851916031626, −0.55607402985922648574029236690, 0.55607402985922648574029236690, 1.57722757377659723851916031626, 2.80454738554761168256656770725, 3.19896036037118686763361609116, 4.72609009512419540977989601285, 4.85704086691754657723695208883, 5.62923163077647156462500404122, 6.19296160655540979528124408319, 7.04616215580093283140604949226, 7.87561251044937135804478764106

Graph of the ZZ-function along the critical line