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Properties

Label 2-9200-1.1-c1-0-101
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  + 1.58·3-s + 2.84·7-s − 0.493·9-s − 1.98·11-s + 4.69·13-s + 3.16·17-s + 6.16·19-s + 4.50·21-s − 23-s − 5.53·27-s − 6.61·29-s + 8.29·31-s − 3.14·33-s − 1.71·37-s + 7.42·39-s + 6.72·41-s + 0.177·43-s − 11.4·47-s + 1.10·49-s + 5.01·51-s + 6.18·53-s + 9.75·57-s + 7.61·59-s + 11.8·61-s − 1.40·63-s − 3.07·67-s − 1.58·69-s + ⋯
L(s)  = 1  + 0.914·3-s + 1.07·7-s − 0.164·9-s − 0.598·11-s + 1.30·13-s + 0.768·17-s + 1.41·19-s + 0.983·21-s − 0.208·23-s − 1.06·27-s − 1.22·29-s + 1.49·31-s − 0.547·33-s − 0.281·37-s + 1.18·39-s + 1.05·41-s + 0.0271·43-s − 1.67·47-s + 0.157·49-s + 0.702·51-s + 0.849·53-s + 1.29·57-s + 0.991·59-s + 1.51·61-s − 0.176·63-s − 0.375·67-s − 0.190·69-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 3.6902257003.690225700
L(12)L(\frac12) ≈\approx 3.6902257003.690225700
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 1−1.58T+3T2 1 - 1.58T + 3T^{2}
7 1−2.84T+7T2 1 - 2.84T + 7T^{2}
11 1+1.98T+11T2 1 + 1.98T + 11T^{2}
13 1−4.69T+13T2 1 - 4.69T + 13T^{2}
17 1−3.16T+17T2 1 - 3.16T + 17T^{2}
19 1−6.16T+19T2 1 - 6.16T + 19T^{2}
29 1+6.61T+29T2 1 + 6.61T + 29T^{2}
31 1−8.29T+31T2 1 - 8.29T + 31T^{2}
37 1+1.71T+37T2 1 + 1.71T + 37T^{2}
41 1−6.72T+41T2 1 - 6.72T + 41T^{2}
43 1−0.177T+43T2 1 - 0.177T + 43T^{2}
47 1+11.4T+47T2 1 + 11.4T + 47T^{2}
53 1−6.18T+53T2 1 - 6.18T + 53T^{2}
59 1−7.61T+59T2 1 - 7.61T + 59T^{2}
61 1−11.8T+61T2 1 - 11.8T + 61T^{2}
67 1+3.07T+67T2 1 + 3.07T + 67T^{2}
71 1−1.96T+71T2 1 - 1.96T + 71T^{2}
73 1−4.94T+73T2 1 - 4.94T + 73T^{2}
79 1+8.53T+79T2 1 + 8.53T + 79T^{2}
83 1+3.49T+83T2 1 + 3.49T + 83T^{2}
89 1−7.07T+89T2 1 - 7.07T + 89T^{2}
97 1+7.00T+97T2 1 + 7.00T + 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

−7.81275533062113652639396639384, −7.39669954172911159674698222310, −6.29557120214045494242116713052, −5.52929469534979614651020603824, −5.08216677488891310996970563657, −4.01398048408968804058778725717, −3.41133082683224528720292833422, −2.68296292652576523796146526101, −1.78265457984193183639139460061, −0.934103500596506240602860314984, 0.934103500596506240602860314984, 1.78265457984193183639139460061, 2.68296292652576523796146526101, 3.41133082683224528720292833422, 4.01398048408968804058778725717, 5.08216677488891310996970563657, 5.52929469534979614651020603824, 6.29557120214045494242116713052, 7.39669954172911159674698222310, 7.81275533062113652639396639384

Graph of the ZZ-function along the critical line