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Properties

Label 2-288-8.3-c10-0-19
Degree 22
Conductor 288288
Sign 11
Analytic cond. 182.982182.982
Root an. cond. 13.527113.5271
Motivic weight 1010
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.74e4·11-s − 8.23e5·17-s − 3.35e6·19-s + 9.76e6·25-s + 3.77e7·41-s − 2.14e8·43-s + 2.82e8·49-s + 9.21e8·59-s − 1.81e9·67-s − 1.60e9·73-s + 9.60e7·83-s + 1.11e10·89-s − 9.87e9·97-s + 2.74e10·107-s − 2.88e10·113-s + ⋯
L(s)  = 1  − 0.604·11-s − 0.580·17-s − 1.35·19-s + 25-s + 0.326·41-s − 1.45·43-s + 49-s + 1.28·59-s − 1.34·67-s − 0.774·73-s + 0.0243·83-s + 1.99·89-s − 1.14·97-s + 1.96·107-s − 1.56·113-s + ⋯

Functional equation

Λ(s)=(288s/2 ΓC(s) L(s)=( Λ(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}
Λ(s)=(288s/2 ΓC(s+5) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 288288    =    25⋅322^{5} \cdot 3^{2}
Sign: 11
Analytic conductor: 182.982182.982
Root analytic conductor: 13.527113.5271
Motivic weight: 1010
Rational: yes
Arithmetic: yes
Character: χ288(271,⋅)\chi_{288} (271, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 288, ( :5), 1)(2,\ 288,\ (\ :5),\ 1)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 1.5397529081.539752908
L(12)L(\frac12) ≈\approx 1.5397529081.539752908
L(6)L(6) 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
3 1 1
good5 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
7 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
11 1+97426T+p10T2 1 + 97426 T + p^{10} T^{2}
13 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
17 1+823682T+p10T2 1 + 823682 T + p^{10} T^{2}
19 1+3353726T+p10T2 1 + 3353726 T + p^{10} T^{2}
23 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
29 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
31 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
37 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
41 1−37778926T+p10T2 1 - 37778926 T + p^{10} T^{2}
43 1+214485614T+p10T2 1 + 214485614 T + p^{10} T^{2}
47 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
53 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
59 1−921043598T+p10T2 1 - 921043598 T + p^{10} T^{2}
61 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
67 1+1813708382T+p10T2 1 + 1813708382 T + p^{10} T^{2}
71 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
73 1+1605781582T+p10T2 1 + 1605781582 T + p^{10} T^{2}
79 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
83 1−96051518T+p10T2 1 - 96051518 T + p^{10} T^{2}
89 1−11116019374T+p10T2 1 - 11116019374 T + p^{10} T^{2}
97 1+9872978014T+p10T2 1 + 9872978014 T + p^{10} T^{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

−10.23174917817097704132387200810, −8.997642362541336893120637068858, −8.284923892782079449622615556105, −7.14465774716899679712754706727, −6.24676004658103149080287814027, −5.09517477600305482506258284604, −4.14799805436231887604603874332, −2.87185188095309228767915767922, −1.88297659535103519488378485087, −0.51343917301733733145802683475, 0.51343917301733733145802683475, 1.88297659535103519488378485087, 2.87185188095309228767915767922, 4.14799805436231887604603874332, 5.09517477600305482506258284604, 6.24676004658103149080287814027, 7.14465774716899679712754706727, 8.284923892782079449622615556105, 8.997642362541336893120637068858, 10.23174917817097704132387200810

Graph of the ZZ-function along the critical line