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Properties

Label 2-2e4-1.1-c27-0-2
Degree 22
Conductor 1616
Sign 11
Analytic cond. 73.896873.8968
Root an. cond. 8.596338.59633
Motivic weight 2727
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.48e5·3-s − 3.54e9·5-s + 2.27e11·7-s − 7.56e12·9-s + 1.01e13·11-s + 6.50e14·13-s − 8.81e14·15-s − 6.16e16·17-s − 2.66e17·19-s + 5.65e16·21-s + 2.43e18·23-s + 5.10e18·25-s − 3.77e18·27-s + 5.02e18·29-s − 2.01e19·31-s + 2.53e18·33-s − 8.05e20·35-s + 1.50e21·37-s + 1.61e20·39-s + 4.94e21·41-s − 1.43e22·43-s + 2.68e22·45-s − 4.74e22·47-s − 1.39e22·49-s − 1.53e22·51-s + 2.37e23·53-s − 3.61e22·55-s + ⋯
L(s)  = 1  + 0.0900·3-s − 1.29·5-s + 0.887·7-s − 0.991·9-s + 0.0890·11-s + 0.595·13-s − 0.116·15-s − 1.51·17-s − 1.45·19-s + 0.0798·21-s + 1.00·23-s + 0.685·25-s − 0.179·27-s + 0.0908·29-s − 0.148·31-s + 0.00802·33-s − 1.15·35-s + 1.01·37-s + 0.0536·39-s + 0.834·41-s − 1.27·43-s + 1.28·45-s − 1.26·47-s − 0.213·49-s − 0.136·51-s + 1.25·53-s − 0.115·55-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=( Λ(28−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+27/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: 11
Analytic conductor: 73.896873.8968
Root analytic conductor: 8.596338.59633
Motivic weight: 2727
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 16, ( :27/2), 1)(2,\ 16,\ (\ :27/2),\ 1)

Particular Values

L(14)L(14) ≈\approx 1.2295925141.229592514
L(12)L(\frac12) ≈\approx 1.2295925141.229592514
L(292)L(\frac{29}{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
good3 1−2.48e5T+7.62e12T2 1 - 2.48e5T + 7.62e12T^{2}
5 1+3.54e9T+7.45e18T2 1 + 3.54e9T + 7.45e18T^{2}
7 1−2.27e11T+6.57e22T2 1 - 2.27e11T + 6.57e22T^{2}
11 1−1.01e13T+1.31e28T2 1 - 1.01e13T + 1.31e28T^{2}
13 1−6.50e14T+1.19e30T2 1 - 6.50e14T + 1.19e30T^{2}
17 1+6.16e16T+1.66e33T2 1 + 6.16e16T + 1.66e33T^{2}
19 1+2.66e17T+3.36e34T2 1 + 2.66e17T + 3.36e34T^{2}
23 1−2.43e18T+5.84e36T2 1 - 2.43e18T + 5.84e36T^{2}
29 1−5.02e18T+3.05e39T2 1 - 5.02e18T + 3.05e39T^{2}
31 1+2.01e19T+1.84e40T2 1 + 2.01e19T + 1.84e40T^{2}
37 1−1.50e21T+2.19e42T2 1 - 1.50e21T + 2.19e42T^{2}
41 1−4.94e21T+3.50e43T2 1 - 4.94e21T + 3.50e43T^{2}
43 1+1.43e22T+1.26e44T2 1 + 1.43e22T + 1.26e44T^{2}
47 1+4.74e22T+1.40e45T2 1 + 4.74e22T + 1.40e45T^{2}
53 1−2.37e23T+3.59e46T2 1 - 2.37e23T + 3.59e46T^{2}
59 1−1.40e24T+6.50e47T2 1 - 1.40e24T + 6.50e47T^{2}
61 1+1.05e24T+1.59e48T2 1 + 1.05e24T + 1.59e48T^{2}
67 1−7.87e24T+2.01e49T2 1 - 7.87e24T + 2.01e49T^{2}
71 1−1.00e25T+9.63e49T2 1 - 1.00e25T + 9.63e49T^{2}
73 1−2.10e25T+2.04e50T2 1 - 2.10e25T + 2.04e50T^{2}
79 1+5.00e25T+1.72e51T2 1 + 5.00e25T + 1.72e51T^{2}
83 1−1.13e26T+6.53e51T2 1 - 1.13e26T + 6.53e51T^{2}
89 1−1.60e26T+4.30e52T2 1 - 1.60e26T + 4.30e52T^{2}
97 1+2.68e26T+4.39e53T2 1 + 2.68e26T + 4.39e53T^{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

−13.06235111946906377698459899186, −11.46186414060994266142064459640, −11.05097098764588178388013438414, −8.750118219323210989983526360573, −8.095224266438183877299494351856, −6.57073315211889655529592969267, −4.82995365379324659246786382869, −3.75495169619207074709089139478, −2.24755530357198709186379764393, −0.55369091138267194760748460350, 0.55369091138267194760748460350, 2.24755530357198709186379764393, 3.75495169619207074709089139478, 4.82995365379324659246786382869, 6.57073315211889655529592969267, 8.095224266438183877299494351856, 8.750118219323210989983526360573, 11.05097098764588178388013438414, 11.46186414060994266142064459640, 13.06235111946906377698459899186

Graph of the ZZ-function along the critical line