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Properties

Label 2-2e4-1.1-c27-0-0
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  − 3.31e6·3-s − 1.51e9·5-s − 7.50e10·7-s + 3.35e12·9-s + 4.82e13·11-s − 1.67e15·13-s + 5.02e15·15-s − 3.32e16·17-s − 9.72e16·19-s + 2.48e17·21-s − 2.52e18·23-s − 5.15e18·25-s + 1.41e19·27-s − 7.47e19·29-s − 1.99e20·31-s − 1.59e20·33-s + 1.13e20·35-s − 1.37e21·37-s + 5.54e21·39-s − 1.75e21·41-s + 1.71e22·43-s − 5.08e21·45-s + 1.00e22·47-s − 6.00e22·49-s + 1.10e23·51-s − 2.26e23·53-s − 7.31e22·55-s + ⋯
L(s)  = 1  − 1.20·3-s − 0.555·5-s − 0.292·7-s + 0.440·9-s + 0.421·11-s − 1.53·13-s + 0.666·15-s − 0.813·17-s − 0.530·19-s + 0.351·21-s − 1.04·23-s − 0.691·25-s + 0.671·27-s − 1.35·29-s − 1.47·31-s − 0.505·33-s + 0.162·35-s − 0.927·37-s + 1.83·39-s − 0.296·41-s + 1.52·43-s − 0.244·45-s + 0.267·47-s − 0.914·49-s + 0.976·51-s − 1.19·53-s − 0.234·55-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 0.094626329470.09462632947
L(12)L(\frac12) ≈\approx 0.094626329470.09462632947
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+3.31e6T+7.62e12T2 1 + 3.31e6T + 7.62e12T^{2}
5 1+1.51e9T+7.45e18T2 1 + 1.51e9T + 7.45e18T^{2}
7 1+7.50e10T+6.57e22T2 1 + 7.50e10T + 6.57e22T^{2}
11 1−4.82e13T+1.31e28T2 1 - 4.82e13T + 1.31e28T^{2}
13 1+1.67e15T+1.19e30T2 1 + 1.67e15T + 1.19e30T^{2}
17 1+3.32e16T+1.66e33T2 1 + 3.32e16T + 1.66e33T^{2}
19 1+9.72e16T+3.36e34T2 1 + 9.72e16T + 3.36e34T^{2}
23 1+2.52e18T+5.84e36T2 1 + 2.52e18T + 5.84e36T^{2}
29 1+7.47e19T+3.05e39T2 1 + 7.47e19T + 3.05e39T^{2}
31 1+1.99e20T+1.84e40T2 1 + 1.99e20T + 1.84e40T^{2}
37 1+1.37e21T+2.19e42T2 1 + 1.37e21T + 2.19e42T^{2}
41 1+1.75e21T+3.50e43T2 1 + 1.75e21T + 3.50e43T^{2}
43 1−1.71e22T+1.26e44T2 1 - 1.71e22T + 1.26e44T^{2}
47 1−1.00e22T+1.40e45T2 1 - 1.00e22T + 1.40e45T^{2}
53 1+2.26e23T+3.59e46T2 1 + 2.26e23T + 3.59e46T^{2}
59 1+1.00e24T+6.50e47T2 1 + 1.00e24T + 6.50e47T^{2}
61 1−1.34e24T+1.59e48T2 1 - 1.34e24T + 1.59e48T^{2}
67 1−5.33e24T+2.01e49T2 1 - 5.33e24T + 2.01e49T^{2}
71 1+7.13e24T+9.63e49T2 1 + 7.13e24T + 9.63e49T^{2}
73 1−6.77e24T+2.04e50T2 1 - 6.77e24T + 2.04e50T^{2}
79 1−2.79e25T+1.72e51T2 1 - 2.79e25T + 1.72e51T^{2}
83 1−9.54e25T+6.53e51T2 1 - 9.54e25T + 6.53e51T^{2}
89 1−3.43e26T+4.30e52T2 1 - 3.43e26T + 4.30e52T^{2}
97 1+4.35e25T+4.39e53T2 1 + 4.35e25T + 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

−12.68555227450607779791960145295, −11.80708479664019110807298842541, −10.78585583532101940849067545807, −9.364386869362460506478533345096, −7.57747903000227396168145152819, −6.37163098535012273445498738032, −5.14138344108970207485010833905, −3.89518578927680416759848005635, −2.05276621156574229601378710517, −0.15910852646769645689039531470, 0.15910852646769645689039531470, 2.05276621156574229601378710517, 3.89518578927680416759848005635, 5.14138344108970207485010833905, 6.37163098535012273445498738032, 7.57747903000227396168145152819, 9.364386869362460506478533345096, 10.78585583532101940849067545807, 11.80708479664019110807298842541, 12.68555227450607779791960145295

Graph of the ZZ-function along the critical line