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Properties

Label 2-91e2-1.1-c1-0-166
Degree 22
Conductor 82818281
Sign −1-1
Analytic cond. 66.124166.1241
Root an. cond. 8.131678.13167
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  + 0.195·2-s − 0.259·3-s − 1.96·4-s − 3.93·5-s − 0.0508·6-s − 0.775·8-s − 2.93·9-s − 0.769·10-s − 4.50·11-s + 0.509·12-s + 1.02·15-s + 3.77·16-s − 2.28·17-s − 0.573·18-s + 1.78·19-s + 7.71·20-s − 0.881·22-s + 1.74·23-s + 0.201·24-s + 10.4·25-s + 1.54·27-s + 1.65·29-s + 0.199·30-s − 5.60·31-s + 2.28·32-s + 1.17·33-s − 0.446·34-s + ⋯
L(s)  = 1  + 0.138·2-s − 0.149·3-s − 0.980·4-s − 1.75·5-s − 0.0207·6-s − 0.274·8-s − 0.977·9-s − 0.243·10-s − 1.35·11-s + 0.147·12-s + 0.263·15-s + 0.942·16-s − 0.553·17-s − 0.135·18-s + 0.410·19-s + 1.72·20-s − 0.187·22-s + 0.362·23-s + 0.0411·24-s + 2.09·25-s + 0.296·27-s + 0.306·29-s + 0.0364·30-s − 1.00·31-s + 0.404·32-s + 0.203·33-s − 0.0765·34-s + ⋯

Functional equation

Λ(s)=(8281s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8281s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 82818281    =    72⋅1327^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 66.124166.1241
Root analytic conductor: 8.131678.13167
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8281, ( :1/2), −1)(2,\ 8281,\ (\ :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)
bad7 1 1
13 1 1
good2 1−0.195T+2T2 1 - 0.195T + 2T^{2}
3 1+0.259T+3T2 1 + 0.259T + 3T^{2}
5 1+3.93T+5T2 1 + 3.93T + 5T^{2}
11 1+4.50T+11T2 1 + 4.50T + 11T^{2}
17 1+2.28T+17T2 1 + 2.28T + 17T^{2}
19 1−1.78T+19T2 1 - 1.78T + 19T^{2}
23 1−1.74T+23T2 1 - 1.74T + 23T^{2}
29 1−1.65T+29T2 1 - 1.65T + 29T^{2}
31 1+5.60T+31T2 1 + 5.60T + 31T^{2}
37 1+7.14T+37T2 1 + 7.14T + 37T^{2}
41 1−8.11T+41T2 1 - 8.11T + 41T^{2}
43 1−6.81T+43T2 1 - 6.81T + 43T^{2}
47 1+3.54T+47T2 1 + 3.54T + 47T^{2}
53 1−3.28T+53T2 1 - 3.28T + 53T^{2}
59 1+4.50T+59T2 1 + 4.50T + 59T^{2}
61 1−7.54T+61T2 1 - 7.54T + 61T^{2}
67 1−12.6T+67T2 1 - 12.6T + 67T^{2}
71 1+9.54T+71T2 1 + 9.54T + 71T^{2}
73 1+1.08T+73T2 1 + 1.08T + 73T^{2}
79 1−0.791T+79T2 1 - 0.791T + 79T^{2}
83 1−7.14T+83T2 1 - 7.14T + 83T^{2}
89 1−11.2T+89T2 1 - 11.2T + 89T^{2}
97 1−8.81T+97T2 1 - 8.81T + 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

−7.69747725115047820594895357667, −6.96182672970872597115185077387, −5.86593503974774212690270397128, −5.17799661148868669093278056813, −4.71177483625010844389279259328, −3.86207666121627903671376796732, −3.30368040025380187539949936712, −2.51816864270507658915657682372, −0.71204636625999536028728804519, 0, 0.71204636625999536028728804519, 2.51816864270507658915657682372, 3.30368040025380187539949936712, 3.86207666121627903671376796732, 4.71177483625010844389279259328, 5.17799661148868669093278056813, 5.86593503974774212690270397128, 6.96182672970872597115185077387, 7.69747725115047820594895357667

Graph of the ZZ-function along the critical line