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Properties

Label 2-9200-1.1-c1-0-5
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  − 1.56·3-s − 3.12·7-s − 0.561·9-s + 4·11-s − 3.56·13-s − 5.12·17-s − 4·19-s + 4.87·21-s + 23-s + 5.56·27-s − 4.43·29-s − 5.56·31-s − 6.24·33-s − 1.12·37-s + 5.56·39-s − 3.56·41-s − 0.876·43-s + 8.68·47-s + 2.75·49-s + 8·51-s − 12.2·53-s + 6.24·57-s − 10.2·59-s + 2.87·61-s + 1.75·63-s − 10.2·67-s − 1.56·69-s + ⋯
L(s)  = 1  − 0.901·3-s − 1.18·7-s − 0.187·9-s + 1.20·11-s − 0.987·13-s − 1.24·17-s − 0.917·19-s + 1.06·21-s + 0.208·23-s + 1.07·27-s − 0.824·29-s − 0.998·31-s − 1.08·33-s − 0.184·37-s + 0.890·39-s − 0.556·41-s − 0.133·43-s + 1.26·47-s + 0.393·49-s + 1.12·51-s − 1.68·53-s + 0.827·57-s − 1.33·59-s + 0.368·61-s + 0.220·63-s − 1.25·67-s − 0.187·69-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 0.27171072580.2717107258
L(12)L(\frac12) ≈\approx 0.27171072580.2717107258
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+1.56T+3T2 1 + 1.56T + 3T^{2}
7 1+3.12T+7T2 1 + 3.12T + 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1+3.56T+13T2 1 + 3.56T + 13T^{2}
17 1+5.12T+17T2 1 + 5.12T + 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
29 1+4.43T+29T2 1 + 4.43T + 29T^{2}
31 1+5.56T+31T2 1 + 5.56T + 31T^{2}
37 1+1.12T+37T2 1 + 1.12T + 37T^{2}
41 1+3.56T+41T2 1 + 3.56T + 41T^{2}
43 1+0.876T+43T2 1 + 0.876T + 43T^{2}
47 1−8.68T+47T2 1 - 8.68T + 47T^{2}
53 1+12.2T+53T2 1 + 12.2T + 53T^{2}
59 1+10.2T+59T2 1 + 10.2T + 59T^{2}
61 1−2.87T+61T2 1 - 2.87T + 61T^{2}
67 1+10.2T+67T2 1 + 10.2T + 67T^{2}
71 1−8.68T+71T2 1 - 8.68T + 71T^{2}
73 1+12.4T+73T2 1 + 12.4T + 73T^{2}
79 1+6.24T+79T2 1 + 6.24T + 79T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+0.246T+97T2 1 + 0.246T + 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

−7.45480139763667597375579683086, −6.80969123593344564895570093493, −6.37136954112793551399311566078, −5.86026382933567027342875817545, −4.96884265983938944240165325104, −4.29880855890434524108311684873, −3.51302689287560635109911466893, −2.64079581489962468221237313991, −1.68042119468169628636225684191, −0.25303825931015543503927748560, 0.25303825931015543503927748560, 1.68042119468169628636225684191, 2.64079581489962468221237313991, 3.51302689287560635109911466893, 4.29880855890434524108311684873, 4.96884265983938944240165325104, 5.86026382933567027342875817545, 6.37136954112793551399311566078, 6.80969123593344564895570093493, 7.45480139763667597375579683086

Graph of the ZZ-function along the critical line