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Properties

Label 2-91e2-1.1-c1-0-137
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  − 0.264·2-s − 2.90·3-s − 1.92·4-s − 1.43·5-s + 0.769·6-s + 1.03·8-s + 5.46·9-s + 0.379·10-s − 5.50·11-s + 5.61·12-s + 4.17·15-s + 3.58·16-s − 4.83·17-s − 1.44·18-s + 2.82·19-s + 2.76·20-s + 1.45·22-s − 5.99·23-s − 3.02·24-s − 2.94·25-s − 7.16·27-s + 1.04·29-s − 1.10·30-s + 9.20·31-s − 3.02·32-s + 16.0·33-s + 1.27·34-s + ⋯
L(s)  = 1  − 0.187·2-s − 1.67·3-s − 0.964·4-s − 0.641·5-s + 0.314·6-s + 0.367·8-s + 1.82·9-s + 0.120·10-s − 1.65·11-s + 1.62·12-s + 1.07·15-s + 0.896·16-s − 1.17·17-s − 0.340·18-s + 0.647·19-s + 0.619·20-s + 0.310·22-s − 1.25·23-s − 0.617·24-s − 0.588·25-s − 1.37·27-s + 0.193·29-s − 0.201·30-s + 1.65·31-s − 0.535·32-s + 2.78·33-s + 0.219·34-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+0.264T+2T2 1 + 0.264T + 2T^{2}
3 1+2.90T+3T2 1 + 2.90T + 3T^{2}
5 1+1.43T+5T2 1 + 1.43T + 5T^{2}
11 1+5.50T+11T2 1 + 5.50T + 11T^{2}
17 1+4.83T+17T2 1 + 4.83T + 17T^{2}
19 1−2.82T+19T2 1 - 2.82T + 19T^{2}
23 1+5.99T+23T2 1 + 5.99T + 23T^{2}
29 1−1.04T+29T2 1 - 1.04T + 29T^{2}
31 1−9.20T+31T2 1 - 9.20T + 31T^{2}
37 1+0.612T+37T2 1 + 0.612T + 37T^{2}
41 1+10.6T+41T2 1 + 10.6T + 41T^{2}
43 1+8.43T+43T2 1 + 8.43T + 43T^{2}
47 1−2.40T+47T2 1 - 2.40T + 47T^{2}
53 1+1.82T+53T2 1 + 1.82T + 53T^{2}
59 1+0.870T+59T2 1 + 0.870T + 59T^{2}
61 1−3.33T+61T2 1 - 3.33T + 61T^{2}
67 1−6.62T+67T2 1 - 6.62T + 67T^{2}
71 1−6.85T+71T2 1 - 6.85T + 71T^{2}
73 1−3.14T+73T2 1 - 3.14T + 73T^{2}
79 1+17.5T+79T2 1 + 17.5T + 79T^{2}
83 1−11.4T+83T2 1 - 11.4T + 83T^{2}
89 1+0.995T+89T2 1 + 0.995T + 89T^{2}
97 1−13.5T+97T2 1 - 13.5T + 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

−7.50397012893604926961555121993, −6.67782096623672558544557225427, −5.97168839791640756789387605397, −5.23226637914084523451683993701, −4.81472412016792942942719278016, −4.23750017548348429970413900191, −3.29611693327061008890604053707, −1.97991923554638624858869646204, −0.64035137803010295417494755200, 0, 0.64035137803010295417494755200, 1.97991923554638624858869646204, 3.29611693327061008890604053707, 4.23750017548348429970413900191, 4.81472412016792942942719278016, 5.23226637914084523451683993701, 5.97168839791640756789387605397, 6.67782096623672558544557225427, 7.50397012893604926961555121993

Graph of the ZZ-function along the critical line