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Properties

Label 2-91e2-1.1-c1-0-501
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  + 2.44·2-s + 0.667·3-s + 3.99·4-s + 0.910·5-s + 1.63·6-s + 4.87·8-s − 2.55·9-s + 2.22·10-s − 3.67·11-s + 2.66·12-s + 0.607·15-s + 3.95·16-s − 7.18·17-s − 6.25·18-s − 1.97·19-s + 3.63·20-s − 9.00·22-s − 0.596·23-s + 3.25·24-s − 4.17·25-s − 3.70·27-s − 3.64·29-s + 1.48·30-s − 7.08·31-s − 0.0786·32-s − 2.45·33-s − 17.5·34-s + ⋯
L(s)  = 1  + 1.73·2-s + 0.385·3-s + 1.99·4-s + 0.407·5-s + 0.667·6-s + 1.72·8-s − 0.851·9-s + 0.704·10-s − 1.10·11-s + 0.769·12-s + 0.156·15-s + 0.987·16-s − 1.74·17-s − 1.47·18-s − 0.453·19-s + 0.812·20-s − 1.91·22-s − 0.124·23-s + 0.664·24-s − 0.834·25-s − 0.713·27-s − 0.677·29-s + 0.271·30-s − 1.27·31-s − 0.0139·32-s − 0.427·33-s − 3.01·34-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 1−2.44T+2T2 1 - 2.44T + 2T^{2}
3 1−0.667T+3T2 1 - 0.667T + 3T^{2}
5 1−0.910T+5T2 1 - 0.910T + 5T^{2}
11 1+3.67T+11T2 1 + 3.67T + 11T^{2}
17 1+7.18T+17T2 1 + 7.18T + 17T^{2}
19 1+1.97T+19T2 1 + 1.97T + 19T^{2}
23 1+0.596T+23T2 1 + 0.596T + 23T^{2}
29 1+3.64T+29T2 1 + 3.64T + 29T^{2}
31 1+7.08T+31T2 1 + 7.08T + 31T^{2}
37 1+0.710T+37T2 1 + 0.710T + 37T^{2}
41 1−5.27T+41T2 1 - 5.27T + 41T^{2}
43 1−11.0T+43T2 1 - 11.0T + 43T^{2}
47 1−12.1T+47T2 1 - 12.1T + 47T^{2}
53 1+11.4T+53T2 1 + 11.4T + 53T^{2}
59 1−9.58T+59T2 1 - 9.58T + 59T^{2}
61 1+6.98T+61T2 1 + 6.98T + 61T^{2}
67 1+1.22T+67T2 1 + 1.22T + 67T^{2}
71 1+11.3T+71T2 1 + 11.3T + 71T^{2}
73 1−6.53T+73T2 1 - 6.53T + 73T^{2}
79 1+11.5T+79T2 1 + 11.5T + 79T^{2}
83 1−7.16T+83T2 1 - 7.16T + 83T^{2}
89 1−12.8T+89T2 1 - 12.8T + 89T^{2}
97 1−9.09T+97T2 1 - 9.09T + 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

−7.36691554493877507484533598355, −6.45072011253946785417754646972, −5.79454652656272370495236158901, −5.49409588041271793853473524632, −4.55187966396588725671304236234, −4.01068794699950245182026093500, −3.15037684518967692120358543407, −2.30281677656626698325721391048, −2.11983104676254876333033922911, 0, 2.11983104676254876333033922911, 2.30281677656626698325721391048, 3.15037684518967692120358543407, 4.01068794699950245182026093500, 4.55187966396588725671304236234, 5.49409588041271793853473524632, 5.79454652656272370495236158901, 6.45072011253946785417754646972, 7.36691554493877507484533598355

Graph of the ZZ-function along the critical line