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Properties

Label 2-9200-1.1-c1-0-126
Degree 22
Conductor 92009200
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.36·3-s − 1.90·7-s + 8.28·9-s + 5.48·11-s + 1.04·13-s + 6.74·17-s + 1.55·19-s + 6.40·21-s − 23-s − 17.7·27-s − 3.38·29-s − 10.9·31-s − 18.4·33-s − 5.26·37-s − 3.52·39-s + 6.09·41-s + 0.403·47-s − 3.36·49-s − 22.6·51-s − 5.88·53-s − 5.20·57-s − 9.60·59-s − 7.09·61-s − 15.8·63-s + 13.7·67-s + 3.36·69-s − 0.478·71-s + ⋯
L(s)  = 1  − 1.93·3-s − 0.720·7-s + 2.76·9-s + 1.65·11-s + 0.291·13-s + 1.63·17-s + 0.355·19-s + 1.39·21-s − 0.208·23-s − 3.42·27-s − 0.628·29-s − 1.96·31-s − 3.20·33-s − 0.865·37-s − 0.564·39-s + 0.952·41-s + 0.0589·47-s − 0.480·49-s − 3.17·51-s − 0.808·53-s − 0.690·57-s − 1.24·59-s − 0.908·61-s − 1.99·63-s + 1.68·67-s + 0.404·69-s − 0.0568·71-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: −1-1
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: 11
Selberg data: (2, 9200, ( :1/2), −1)(2,\ 9200,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+3.36T+3T2 1 + 3.36T + 3T^{2}
7 1+1.90T+7T2 1 + 1.90T + 7T^{2}
11 1−5.48T+11T2 1 - 5.48T + 11T^{2}
13 1−1.04T+13T2 1 - 1.04T + 13T^{2}
17 1−6.74T+17T2 1 - 6.74T + 17T^{2}
19 1−1.55T+19T2 1 - 1.55T + 19T^{2}
29 1+3.38T+29T2 1 + 3.38T + 29T^{2}
31 1+10.9T+31T2 1 + 10.9T + 31T^{2}
37 1+5.26T+37T2 1 + 5.26T + 37T^{2}
41 1−6.09T+41T2 1 - 6.09T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1−0.403T+47T2 1 - 0.403T + 47T^{2}
53 1+5.88T+53T2 1 + 5.88T + 53T^{2}
59 1+9.60T+59T2 1 + 9.60T + 59T^{2}
61 1+7.09T+61T2 1 + 7.09T + 61T^{2}
67 1−13.7T+67T2 1 - 13.7T + 67T^{2}
71 1+0.478T+71T2 1 + 0.478T + 71T^{2}
73 1+2.40T+73T2 1 + 2.40T + 73T^{2}
79 1−4.24T+79T2 1 - 4.24T + 79T^{2}
83 1+11.2T+83T2 1 + 11.2T + 83T^{2}
89 1−4.90T+89T2 1 - 4.90T + 89T^{2}
97 1−12.3T+97T2 1 - 12.3T + 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

−7.21728027149189086301873797378, −6.47965577836181310331533219791, −6.01224322772566753564456047110, −5.51930549596196555428905770723, −4.77316857366563257395799156557, −3.80123525488652340368685471074, −3.49282828780944601599920988642, −1.68188144223618512726653471256, −1.09893989760303568293088236461, 0, 1.09893989760303568293088236461, 1.68188144223618512726653471256, 3.49282828780944601599920988642, 3.80123525488652340368685471074, 4.77316857366563257395799156557, 5.51930549596196555428905770723, 6.01224322772566753564456047110, 6.47965577836181310331533219791, 7.21728027149189086301873797378

Graph of the ZZ-function along the critical line