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Properties

Label 2-91e2-1.1-c1-0-486
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  + 2.18·2-s + 1.79·3-s + 2.79·4-s − 2.18·5-s + 3.92·6-s + 1.73·8-s + 0.208·9-s − 4.79·10-s + 1.27·11-s + 4.99·12-s − 3.92·15-s − 1.79·16-s + 3·17-s + 0.456·18-s − 6.56·19-s − 6.10·20-s + 2.79·22-s − 7.58·23-s + 3.10·24-s − 0.208·25-s − 5.00·27-s − 2.20·29-s − 8.58·30-s + 8.66·31-s − 7.38·32-s + 2.28·33-s + 6.56·34-s + ⋯
L(s)  = 1  + 1.54·2-s + 1.03·3-s + 1.39·4-s − 0.978·5-s + 1.60·6-s + 0.612·8-s + 0.0695·9-s − 1.51·10-s + 0.384·11-s + 1.44·12-s − 1.01·15-s − 0.447·16-s + 0.727·17-s + 0.107·18-s − 1.50·19-s − 1.36·20-s + 0.595·22-s − 1.58·23-s + 0.633·24-s − 0.0417·25-s − 0.962·27-s − 0.410·29-s − 1.56·30-s + 1.55·31-s − 1.30·32-s + 0.397·33-s + 1.12·34-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 1−2.18T+2T2 1 - 2.18T + 2T^{2}
3 1−1.79T+3T2 1 - 1.79T + 3T^{2}
5 1+2.18T+5T2 1 + 2.18T + 5T^{2}
11 1−1.27T+11T2 1 - 1.27T + 11T^{2}
17 1−3T+17T2 1 - 3T + 17T^{2}
19 1+6.56T+19T2 1 + 6.56T + 19T^{2}
23 1+7.58T+23T2 1 + 7.58T + 23T^{2}
29 1+2.20T+29T2 1 + 2.20T + 29T^{2}
31 1−8.66T+31T2 1 - 8.66T + 31T^{2}
37 1+6.92T+37T2 1 + 6.92T + 37T^{2}
41 1+2.55T+41T2 1 + 2.55T + 41T^{2}
43 1−4.37T+43T2 1 - 4.37T + 43T^{2}
47 1−4.28T+47T2 1 - 4.28T + 47T^{2}
53 1+12.1T+53T2 1 + 12.1T + 53T^{2}
59 1−8.85T+59T2 1 - 8.85T + 59T^{2}
61 1+12.7T+61T2 1 + 12.7T + 61T^{2}
67 1−11.4T+67T2 1 - 11.4T + 67T^{2}
71 1−0.913T+71T2 1 - 0.913T + 71T^{2}
73 1−3.46T+73T2 1 - 3.46T + 73T^{2}
79 1+6T+79T2 1 + 6T + 79T^{2}
83 1−3.55T+83T2 1 - 3.55T + 83T^{2}
89 1+2.91T+89T2 1 + 2.91T + 89T^{2}
97 1+15.2T+97T2 1 + 15.2T + 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

−7.47634591908627826580838616126, −6.57233324901919558392604121138, −6.05311358179915019128681018624, −5.22648955931779422700274267481, −4.23852992229881899054544169637, −3.98795208474836277896977023973, −3.33579453188552984446874310907, −2.58871452474296397847341028630, −1.81787943167758195094476632093, 0, 1.81787943167758195094476632093, 2.58871452474296397847341028630, 3.33579453188552984446874310907, 3.98795208474836277896977023973, 4.23852992229881899054544169637, 5.22648955931779422700274267481, 6.05311358179915019128681018624, 6.57233324901919558392604121138, 7.47634591908627826580838616126

Graph of the ZZ-function along the critical line