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Properties

Label 2-1620-1.1-c1-0-12
Degree 22
Conductor 16201620
Sign −1-1
Analytic cond. 12.935712.9357
Root an. cond. 3.596633.59663
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  − 5-s + 0.732·7-s − 1.73·11-s − 1.46·13-s − 1.26·17-s + 2.46·19-s − 3.46·23-s + 25-s − 4.26·29-s − 7.92·31-s − 0.732·35-s + 4.19·37-s − 0.803·41-s + 6.73·43-s − 4.73·47-s − 6.46·49-s − 10.7·53-s + 1.73·55-s − 4.26·59-s − 4·61-s + 1.46·65-s − 14.3·67-s − 0.803·71-s + 10.1·73-s − 1.26·77-s + 6.39·79-s + 9.12·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.276·7-s − 0.522·11-s − 0.406·13-s − 0.307·17-s + 0.565·19-s − 0.722·23-s + 0.200·25-s − 0.792·29-s − 1.42·31-s − 0.123·35-s + 0.689·37-s − 0.125·41-s + 1.02·43-s − 0.690·47-s − 0.923·49-s − 1.47·53-s + 0.233·55-s − 0.555·59-s − 0.512·61-s + 0.181·65-s − 1.75·67-s − 0.0953·71-s + 1.19·73-s − 0.144·77-s + 0.719·79-s + 1.00·83-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: −1-1
Analytic conductor: 12.935712.9357
Root analytic conductor: 3.596633.59663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1620, ( :1/2), −1)(2,\ 1620,\ (\ :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)
bad2 1 1
3 1 1
5 1+T 1 + T
good7 1−0.732T+7T2 1 - 0.732T + 7T^{2}
11 1+1.73T+11T2 1 + 1.73T + 11T^{2}
13 1+1.46T+13T2 1 + 1.46T + 13T^{2}
17 1+1.26T+17T2 1 + 1.26T + 17T^{2}
19 1−2.46T+19T2 1 - 2.46T + 19T^{2}
23 1+3.46T+23T2 1 + 3.46T + 23T^{2}
29 1+4.26T+29T2 1 + 4.26T + 29T^{2}
31 1+7.92T+31T2 1 + 7.92T + 31T^{2}
37 1−4.19T+37T2 1 - 4.19T + 37T^{2}
41 1+0.803T+41T2 1 + 0.803T + 41T^{2}
43 1−6.73T+43T2 1 - 6.73T + 43T^{2}
47 1+4.73T+47T2 1 + 4.73T + 47T^{2}
53 1+10.7T+53T2 1 + 10.7T + 53T^{2}
59 1+4.26T+59T2 1 + 4.26T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1+14.3T+67T2 1 + 14.3T + 67T^{2}
71 1+0.803T+71T2 1 + 0.803T + 71T^{2}
73 1−10.1T+73T2 1 - 10.1T + 73T^{2}
79 1−6.39T+79T2 1 - 6.39T + 79T^{2}
83 1−9.12T+83T2 1 - 9.12T + 83T^{2}
89 1+5.19T+89T2 1 + 5.19T + 89T^{2}
97 1+2.73T+97T2 1 + 2.73T + 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

−9.110609864033189596256480630968, −7.900682435361490441100623140208, −7.69392605497610704803702373014, −6.62949699463280974395363368609, −5.64426769097997952436996383696, −4.84235530744126004225656247974, −3.92384276627848462930723316495, −2.90305631375347707399054116567, −1.71908120656316357702035224078, 0, 1.71908120656316357702035224078, 2.90305631375347707399054116567, 3.92384276627848462930723316495, 4.84235530744126004225656247974, 5.64426769097997952436996383696, 6.62949699463280974395363368609, 7.69392605497610704803702373014, 7.900682435361490441100623140208, 9.110609864033189596256480630968

Graph of the ZZ-function along the critical line