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Properties

Label 2-91e2-1.1-c1-0-33
Degree 22
Conductor 82818281
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.34·2-s + 1.14·3-s + 3.48·4-s − 1.34·5-s − 2.68·6-s − 3.48·8-s − 1.68·9-s + 3.14·10-s − 1.14·11-s + 4.00·12-s − 1.53·15-s + 1.19·16-s − 5.83·17-s + 3.94·18-s − 3.34·19-s − 4.68·20-s + 2.68·22-s − 3.17·23-s − 4.00·24-s − 3.19·25-s − 5.37·27-s + 10.4·29-s + 3.60·30-s + 1.63·31-s + 4.17·32-s − 1.31·33-s + 13.6·34-s + ⋯
L(s)  = 1  − 1.65·2-s + 0.661·3-s + 1.74·4-s − 0.600·5-s − 1.09·6-s − 1.23·8-s − 0.561·9-s + 0.994·10-s − 0.345·11-s + 1.15·12-s − 0.397·15-s + 0.299·16-s − 1.41·17-s + 0.930·18-s − 0.766·19-s − 1.04·20-s + 0.572·22-s − 0.662·23-s − 0.816·24-s − 0.639·25-s − 1.03·27-s + 1.94·29-s + 0.658·30-s + 0.293·31-s + 0.738·32-s − 0.228·33-s + 2.34·34-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: 11
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: 00
Selberg data: (2, 8281, ( :1/2), 1)(2,\ 8281,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.29466885950.2946688595
L(12)L(\frac12) ≈\approx 0.29466885950.2946688595
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+2.34T+2T2 1 + 2.34T + 2T^{2}
3 1−1.14T+3T2 1 - 1.14T + 3T^{2}
5 1+1.34T+5T2 1 + 1.34T + 5T^{2}
11 1+1.14T+11T2 1 + 1.14T + 11T^{2}
17 1+5.83T+17T2 1 + 5.83T + 17T^{2}
19 1+3.34T+19T2 1 + 3.34T + 19T^{2}
23 1+3.17T+23T2 1 + 3.17T + 23T^{2}
29 1−10.4T+29T2 1 - 10.4T + 29T^{2}
31 1−1.63T+31T2 1 - 1.63T + 31T^{2}
37 1+8.51T+37T2 1 + 8.51T + 37T^{2}
41 1+0.292T+41T2 1 + 0.292T + 41T^{2}
43 1+8.15T+43T2 1 + 8.15T + 43T^{2}
47 1+10.6T+47T2 1 + 10.6T + 47T^{2}
53 1+0.782T+53T2 1 + 0.782T + 53T^{2}
59 1−12.6T+59T2 1 - 12.6T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−6.10T+67T2 1 - 6.10T + 67T^{2}
71 1+1.53T+71T2 1 + 1.53T + 71T^{2}
73 1+15.3T+73T2 1 + 15.3T + 73T^{2}
79 1−0.882T+79T2 1 - 0.882T + 79T^{2}
83 1+12.1T+83T2 1 + 12.1T + 83T^{2}
89 1−5.73T+89T2 1 - 5.73T + 89T^{2}
97 1+5.34T+97T2 1 + 5.34T + 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

−8.218468652043803442478317019559, −7.37211938999668711485510974266, −6.72813126981089345010888097907, −6.15553634445176828404268341318, −4.95430035750646220004965193425, −4.14028191921784606671357751026, −3.16595504086723249201992777847, −2.38075751983660500864425938109, −1.74110741684718961817209880768, −0.31546637797499390687043156064, 0.31546637797499390687043156064, 1.74110741684718961817209880768, 2.38075751983660500864425938109, 3.16595504086723249201992777847, 4.14028191921784606671357751026, 4.95430035750646220004965193425, 6.15553634445176828404268341318, 6.72813126981089345010888097907, 7.37211938999668711485510974266, 8.218468652043803442478317019559

Graph of the ZZ-function along the critical line