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Properties

Label 2-9792-1.1-c1-0-62
Degree 22
Conductor 97929792
Sign −1-1
Analytic cond. 78.189578.1895
Root an. cond. 8.842488.84248
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  − 3.46·5-s − 2.73·7-s − 1.26·11-s − 5.46·13-s + 17-s + 1.46·19-s + 1.26·23-s + 6.99·25-s + 3.46·29-s + 4.19·31-s + 9.46·35-s − 4.53·37-s + 6·41-s + 8.39·43-s + 6.92·47-s + 0.464·49-s + 12.9·53-s + 4.39·55-s + 2.53·59-s + 0.535·61-s + 18.9·65-s − 14.9·67-s + 8.19·71-s + 2·73-s + 3.46·77-s − 12.1·79-s − 2.53·83-s + ⋯
L(s)  = 1  − 1.54·5-s − 1.03·7-s − 0.382·11-s − 1.51·13-s + 0.242·17-s + 0.335·19-s + 0.264·23-s + 1.39·25-s + 0.643·29-s + 0.753·31-s + 1.59·35-s − 0.745·37-s + 0.937·41-s + 1.27·43-s + 1.01·47-s + 0.0663·49-s + 1.77·53-s + 0.592·55-s + 0.330·59-s + 0.0686·61-s + 2.34·65-s − 1.82·67-s + 0.972·71-s + 0.234·73-s + 0.394·77-s − 1.37·79-s − 0.278·83-s + ⋯

Functional equation

Λ(s)=(9792s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9792s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 97929792    =    26⋅32⋅172^{6} \cdot 3^{2} \cdot 17
Sign: −1-1
Analytic conductor: 78.189578.1895
Root analytic conductor: 8.842488.84248
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9792, ( :1/2), −1)(2,\ 9792,\ (\ :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
17 1−T 1 - T
good5 1+3.46T+5T2 1 + 3.46T + 5T^{2}
7 1+2.73T+7T2 1 + 2.73T + 7T^{2}
11 1+1.26T+11T2 1 + 1.26T + 11T^{2}
13 1+5.46T+13T2 1 + 5.46T + 13T^{2}
19 1−1.46T+19T2 1 - 1.46T + 19T^{2}
23 1−1.26T+23T2 1 - 1.26T + 23T^{2}
29 1−3.46T+29T2 1 - 3.46T + 29T^{2}
31 1−4.19T+31T2 1 - 4.19T + 31T^{2}
37 1+4.53T+37T2 1 + 4.53T + 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1−8.39T+43T2 1 - 8.39T + 43T^{2}
47 1−6.92T+47T2 1 - 6.92T + 47T^{2}
53 1−12.9T+53T2 1 - 12.9T + 53T^{2}
59 1−2.53T+59T2 1 - 2.53T + 59T^{2}
61 1−0.535T+61T2 1 - 0.535T + 61T^{2}
67 1+14.9T+67T2 1 + 14.9T + 67T^{2}
71 1−8.19T+71T2 1 - 8.19T + 71T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1+12.1T+79T2 1 + 12.1T + 79T^{2}
83 1+2.53T+83T2 1 + 2.53T + 83T^{2}
89 1+2.53T+89T2 1 + 2.53T + 89T^{2}
97 1+4.92T+97T2 1 + 4.92T + 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

−7.28002282101717724230790324787, −6.97854172489122686917638755212, −5.98126657208506740057739226688, −5.19028251777974610554366868457, −4.42198203211738445711952393374, −3.86448379631288230893354718176, −2.96207482245245478389092757692, −2.54295528130151363942129121755, −0.850133744635735652625027308449, 0, 0.850133744635735652625027308449, 2.54295528130151363942129121755, 2.96207482245245478389092757692, 3.86448379631288230893354718176, 4.42198203211738445711952393374, 5.19028251777974610554366868457, 5.98126657208506740057739226688, 6.97854172489122686917638755212, 7.28002282101717724230790324787

Graph of the ZZ-function along the critical line