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Properties

Label 2-91e2-1.1-c1-0-293
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  − 1.52·2-s − 2.12·3-s + 0.312·4-s + 0.589·5-s + 3.23·6-s + 2.56·8-s + 1.52·9-s − 0.896·10-s + 1.52·11-s − 0.664·12-s − 1.25·15-s − 4.52·16-s + 4.79·17-s − 2.31·18-s + 1.68·19-s + 0.184·20-s − 2.31·22-s + 1.77·23-s − 5.45·24-s − 4.65·25-s + 3.14·27-s + 6.89·29-s + 1.90·30-s − 6.08·31-s + 1.75·32-s − 3.23·33-s − 7.29·34-s + ⋯
L(s)  = 1  − 1.07·2-s − 1.22·3-s + 0.156·4-s + 0.263·5-s + 1.32·6-s + 0.907·8-s + 0.506·9-s − 0.283·10-s + 0.458·11-s − 0.191·12-s − 0.323·15-s − 1.13·16-s + 1.16·17-s − 0.545·18-s + 0.386·19-s + 0.0412·20-s − 0.493·22-s + 0.369·23-s − 1.11·24-s − 0.930·25-s + 0.605·27-s + 1.27·29-s + 0.347·30-s − 1.09·31-s + 0.310·32-s − 0.562·33-s − 1.25·34-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+1.52T+2T2 1 + 1.52T + 2T^{2}
3 1+2.12T+3T2 1 + 2.12T + 3T^{2}
5 1−0.589T+5T2 1 - 0.589T + 5T^{2}
11 1−1.52T+11T2 1 - 1.52T + 11T^{2}
17 1−4.79T+17T2 1 - 4.79T + 17T^{2}
19 1−1.68T+19T2 1 - 1.68T + 19T^{2}
23 1−1.77T+23T2 1 - 1.77T + 23T^{2}
29 1−6.89T+29T2 1 - 6.89T + 29T^{2}
31 1+6.08T+31T2 1 + 6.08T + 31T^{2}
37 1−1.40T+37T2 1 - 1.40T + 37T^{2}
41 1−1.35T+41T2 1 - 1.35T + 41T^{2}
43 1+11.5T+43T2 1 + 11.5T + 43T^{2}
47 1+0.464T+47T2 1 + 0.464T + 47T^{2}
53 1−8.24T+53T2 1 - 8.24T + 53T^{2}
59 1+11.8T+59T2 1 + 11.8T + 59T^{2}
61 1−2.48T+61T2 1 - 2.48T + 61T^{2}
67 1+7.57T+67T2 1 + 7.57T + 67T^{2}
71 1−6.60T+71T2 1 - 6.60T + 71T^{2}
73 1+16.3T+73T2 1 + 16.3T + 73T^{2}
79 1+14.9T+79T2 1 + 14.9T + 79T^{2}
83 1−10.1T+83T2 1 - 10.1T + 83T^{2}
89 1−16.4T+89T2 1 - 16.4T + 89T^{2}
97 1+0.973T+97T2 1 + 0.973T + 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

−7.51116789650292935273687331545, −6.83251816991747628302390376988, −6.12850128810339044300917925846, −5.43938099489233387725037613235, −4.87547424252547275722417765539, −4.01969848250779287701935826843, −2.99914298368258158543816200282, −1.67159926235190607789738035251, −1.00658413089117427348390392908, 0, 1.00658413089117427348390392908, 1.67159926235190607789738035251, 2.99914298368258158543816200282, 4.01969848250779287701935826843, 4.87547424252547275722417765539, 5.43938099489233387725037613235, 6.12850128810339044300917925846, 6.83251816991747628302390376988, 7.51116789650292935273687331545

Graph of the ZZ-function along the critical line