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Properties

Label 2-9280-1.1-c1-0-125
Degree 22
Conductor 92809280
Sign −1-1
Analytic cond. 74.101174.1011
Root an. cond. 8.608208.60820
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  − 2.93·3-s + 5-s − 1.25·7-s + 5.61·9-s − 2.50·11-s + 2.93·13-s − 2.93·15-s + 7.12·17-s − 4.85·19-s + 3.68·21-s + 1.57·23-s + 25-s − 7.68·27-s + 29-s − 4.61·31-s + 7.36·33-s − 1.25·35-s − 9.87·37-s − 8.61·39-s + 0.508·41-s − 1.38·43-s + 5.61·45-s − 1.36·47-s − 5.42·49-s − 20.9·51-s − 2.23·53-s − 2.50·55-s + ⋯
L(s)  = 1  − 1.69·3-s + 0.447·5-s − 0.474·7-s + 1.87·9-s − 0.756·11-s + 0.814·13-s − 0.757·15-s + 1.72·17-s − 1.11·19-s + 0.803·21-s + 0.327·23-s + 0.200·25-s − 1.47·27-s + 0.185·29-s − 0.829·31-s + 1.28·33-s − 0.211·35-s − 1.62·37-s − 1.37·39-s + 0.0793·41-s − 0.210·43-s + 0.837·45-s − 0.198·47-s − 0.775·49-s − 2.92·51-s − 0.307·53-s − 0.338·55-s + ⋯

Functional equation

Λ(s)=(9280s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9280s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92809280    =    26⋅5⋅292^{6} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 74.101174.1011
Root analytic conductor: 8.608208.60820
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9280, ( :1/2), −1)(2,\ 9280,\ (\ :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−T 1 - T
29 1−T 1 - T
good3 1+2.93T+3T2 1 + 2.93T + 3T^{2}
7 1+1.25T+7T2 1 + 1.25T + 7T^{2}
11 1+2.50T+11T2 1 + 2.50T + 11T^{2}
13 1−2.93T+13T2 1 - 2.93T + 13T^{2}
17 1−7.12T+17T2 1 - 7.12T + 17T^{2}
19 1+4.85T+19T2 1 + 4.85T + 19T^{2}
23 1−1.57T+23T2 1 - 1.57T + 23T^{2}
31 1+4.61T+31T2 1 + 4.61T + 31T^{2}
37 1+9.87T+37T2 1 + 9.87T + 37T^{2}
41 1−0.508T+41T2 1 - 0.508T + 41T^{2}
43 1+1.38T+43T2 1 + 1.38T + 43T^{2}
47 1+1.36T+47T2 1 + 1.36T + 47T^{2}
53 1+2.23T+53T2 1 + 2.23T + 53T^{2}
59 1−11.6T+59T2 1 - 11.6T + 59T^{2}
61 1+3.41T+61T2 1 + 3.41T + 61T^{2}
67 1−6.72T+67T2 1 - 6.72T + 67T^{2}
71 1−14.7T+71T2 1 - 14.7T + 71T^{2}
73 1+12.3T+73T2 1 + 12.3T + 73T^{2}
79 1−3.91T+79T2 1 - 3.91T + 79T^{2}
83 1−1.87T+83T2 1 - 1.87T + 83T^{2}
89 1+11.8T+89T2 1 + 11.8T + 89T^{2}
97 1−5.31T+97T2 1 - 5.31T + 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

−7.06931705533546717400160870790, −6.56311255125387578210757106787, −5.91209017111861440109587376267, −5.40638335675322878297960812709, −4.94924088415403238007292198714, −3.90830726588725094383852845775, −3.17684655045528539998396203416, −1.91974995522407848572741257978, −1.01682144007581638035205575672, 0, 1.01682144007581638035205575672, 1.91974995522407848572741257978, 3.17684655045528539998396203416, 3.90830726588725094383852845775, 4.94924088415403238007292198714, 5.40638335675322878297960812709, 5.91209017111861440109587376267, 6.56311255125387578210757106787, 7.06931705533546717400160870790

Graph of the ZZ-function along the critical line