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Properties

Label 2-1466-1.1-c1-0-11
Degree 22
Conductor 14661466
Sign 11
Analytic cond. 11.706011.7060
Root an. cond. 3.421413.42141
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-s − 2.76·3-s + 4-s − 2.37·5-s − 2.76·6-s + 3.21·7-s + 8-s + 4.65·9-s − 2.37·10-s + 4.20·11-s − 2.76·12-s − 5.77·13-s + 3.21·14-s + 6.57·15-s + 16-s − 0.535·17-s + 4.65·18-s − 7.30·19-s − 2.37·20-s − 8.89·21-s + 4.20·22-s + 0.874·23-s − 2.76·24-s + 0.647·25-s − 5.77·26-s − 4.59·27-s + 3.21·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.59·3-s + 0.5·4-s − 1.06·5-s − 1.12·6-s + 1.21·7-s + 0.353·8-s + 1.55·9-s − 0.751·10-s + 1.26·11-s − 0.798·12-s − 1.60·13-s + 0.858·14-s + 1.69·15-s + 0.250·16-s − 0.129·17-s + 1.09·18-s − 1.67·19-s − 0.531·20-s − 1.94·21-s + 0.895·22-s + 0.182·23-s − 0.564·24-s + 0.129·25-s − 1.13·26-s − 0.883·27-s + 0.607·28-s + ⋯

Functional equation

Λ(s)=(1466s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1466s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14661466    =    2⋅7332 \cdot 733
Sign: 11
Analytic conductor: 11.706011.7060
Root analytic conductor: 3.421413.42141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1466, ( :1/2), 1)(2,\ 1466,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3346961751.334696175
L(12)L(\frac12) ≈\approx 1.3346961751.334696175
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)
bad2 1−T 1 - T
733 1+T 1 + T
good3 1+2.76T+3T2 1 + 2.76T + 3T^{2}
5 1+2.37T+5T2 1 + 2.37T + 5T^{2}
7 1−3.21T+7T2 1 - 3.21T + 7T^{2}
11 1−4.20T+11T2 1 - 4.20T + 11T^{2}
13 1+5.77T+13T2 1 + 5.77T + 13T^{2}
17 1+0.535T+17T2 1 + 0.535T + 17T^{2}
19 1+7.30T+19T2 1 + 7.30T + 19T^{2}
23 1−0.874T+23T2 1 - 0.874T + 23T^{2}
29 1−3.90T+29T2 1 - 3.90T + 29T^{2}
31 1−9.16T+31T2 1 - 9.16T + 31T^{2}
37 1−1.62T+37T2 1 - 1.62T + 37T^{2}
41 1−5.93T+41T2 1 - 5.93T + 41T^{2}
43 1−9.62T+43T2 1 - 9.62T + 43T^{2}
47 1+1.80T+47T2 1 + 1.80T + 47T^{2}
53 1+3.26T+53T2 1 + 3.26T + 53T^{2}
59 1−3.66T+59T2 1 - 3.66T + 59T^{2}
61 1−9.94T+61T2 1 - 9.94T + 61T^{2}
67 1−5.32T+67T2 1 - 5.32T + 67T^{2}
71 1−14.5T+71T2 1 - 14.5T + 71T^{2}
73 1−14.7T+73T2 1 - 14.7T + 73T^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1+15.3T+83T2 1 + 15.3T + 83T^{2}
89 1−9.71T+89T2 1 - 9.71T + 89T^{2}
97 1−14.3T+97T2 1 - 14.3T + 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

−9.804404874368668367171563720777, −8.471113613801767941076497913829, −7.67104856693367987289547687734, −6.81773243105930187543601036375, −6.24521774722744571497312932441, −5.12932025066148753398153992920, −4.48763794189486312521177870630, −4.11526676491209406648488947478, −2.29342541528188133440289867015, −0.814293367361172536067878162554, 0.814293367361172536067878162554, 2.29342541528188133440289867015, 4.11526676491209406648488947478, 4.48763794189486312521177870630, 5.12932025066148753398153992920, 6.24521774722744571497312932441, 6.81773243105930187543601036375, 7.67104856693367987289547687734, 8.471113613801767941076497913829, 9.804404874368668367171563720777

Graph of the ZZ-function along the critical line