↑

Properties

Label 2-2e4-1.1-c27-0-11
Degree 22
Conductor 1616
Sign −1-1
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 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.00e6·3-s + 9.02e8·5-s − 3.76e11·7-s + 1.42e12·9-s + 1.19e14·11-s + 8.45e14·13-s + 2.71e15·15-s − 5.80e16·17-s + 2.28e16·19-s − 1.13e18·21-s + 4.11e18·23-s − 6.63e18·25-s − 1.86e19·27-s − 5.09e19·29-s − 2.28e20·31-s + 3.60e20·33-s − 3.40e20·35-s − 1.37e21·37-s + 2.54e21·39-s + 1.26e21·41-s + 1.57e22·43-s + 1.28e21·45-s − 1.74e22·47-s + 7.64e22·49-s − 1.74e23·51-s − 2.42e23·53-s + 1.08e23·55-s + ⋯
L(s)  = 1  + 1.08·3-s + 0.330·5-s − 1.47·7-s + 0.186·9-s + 1.04·11-s + 0.774·13-s + 0.360·15-s − 1.42·17-s + 0.124·19-s − 1.60·21-s + 1.70·23-s − 0.890·25-s − 0.885·27-s − 0.921·29-s − 1.68·31-s + 1.14·33-s − 0.486·35-s − 0.925·37-s + 0.843·39-s + 0.212·41-s + 1.40·43-s + 0.0617·45-s − 0.465·47-s + 1.16·49-s − 1.54·51-s − 1.28·53-s + 0.346·55-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: −1-1
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: 11
Selberg data: (2, 16, ( :27/2), −1)(2,\ 16,\ (\ :27/2),\ -1)

Particular Values

L(14)L(14) == 00
L(12)L(\frac12) == 00
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 1−3.00e6T+7.62e12T2 1 - 3.00e6T + 7.62e12T^{2}
5 1−9.02e8T+7.45e18T2 1 - 9.02e8T + 7.45e18T^{2}
7 1+3.76e11T+6.57e22T2 1 + 3.76e11T + 6.57e22T^{2}
11 1−1.19e14T+1.31e28T2 1 - 1.19e14T + 1.31e28T^{2}
13 1−8.45e14T+1.19e30T2 1 - 8.45e14T + 1.19e30T^{2}
17 1+5.80e16T+1.66e33T2 1 + 5.80e16T + 1.66e33T^{2}
19 1−2.28e16T+3.36e34T2 1 - 2.28e16T + 3.36e34T^{2}
23 1−4.11e18T+5.84e36T2 1 - 4.11e18T + 5.84e36T^{2}
29 1+5.09e19T+3.05e39T2 1 + 5.09e19T + 3.05e39T^{2}
31 1+2.28e20T+1.84e40T2 1 + 2.28e20T + 1.84e40T^{2}
37 1+1.37e21T+2.19e42T2 1 + 1.37e21T + 2.19e42T^{2}
41 1−1.26e21T+3.50e43T2 1 - 1.26e21T + 3.50e43T^{2}
43 1−1.57e22T+1.26e44T2 1 - 1.57e22T + 1.26e44T^{2}
47 1+1.74e22T+1.40e45T2 1 + 1.74e22T + 1.40e45T^{2}
53 1+2.42e23T+3.59e46T2 1 + 2.42e23T + 3.59e46T^{2}
59 1−4.33e23T+6.50e47T2 1 - 4.33e23T + 6.50e47T^{2}
61 1+1.61e24T+1.59e48T2 1 + 1.61e24T + 1.59e48T^{2}
67 1+3.04e24T+2.01e49T2 1 + 3.04e24T + 2.01e49T^{2}
71 1−1.33e24T+9.63e49T2 1 - 1.33e24T + 9.63e49T^{2}
73 1+2.26e25T+2.04e50T2 1 + 2.26e25T + 2.04e50T^{2}
79 1+2.50e25T+1.72e51T2 1 + 2.50e25T + 1.72e51T^{2}
83 1+2.43e25T+6.53e51T2 1 + 2.43e25T + 6.53e51T^{2}
89 1−3.08e26T+4.30e52T2 1 - 3.08e26T + 4.30e52T^{2}
97 1+2.50e26T+4.39e53T2 1 + 2.50e26T + 4.39e53T^{2}
show more
show less
   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

−12.90125399241009486083222237024, −11.08038271972932137103011784095, −9.299083432650613057169076553401, −8.997511905813082241747603891968, −7.12026981783871185705887621263, −5.99495918612327610638633970875, −3.88570679795934219877029046643, −3.02663056496524052145031640765, −1.71502809173221752353720406658, 0, 1.71502809173221752353720406658, 3.02663056496524052145031640765, 3.88570679795934219877029046643, 5.99495918612327610638633970875, 7.12026981783871185705887621263, 8.997511905813082241747603891968, 9.299083432650613057169076553401, 11.08038271972932137103011784095, 12.90125399241009486083222237024

Graph of the ZZ-function along the critical line