↑

Properties

Label 2-20400-1.1-c1-0-42
Degree 22
Conductor 2040020400
Sign 11
Analytic cond. 162.894162.894
Root an. cond. 12.763012.7630
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·7-s + 9-s + 3·11-s − 4·13-s − 17-s + 5·19-s + 3·21-s + 4·23-s + 27-s − 7·31-s + 3·33-s − 3·37-s − 4·39-s + 2·41-s − 43-s + 3·47-s + 2·49-s − 51-s + 11·53-s + 5·57-s + 2·61-s + 3·63-s + 13·67-s + 4·69-s − 2·71-s + 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 0.242·17-s + 1.14·19-s + 0.654·21-s + 0.834·23-s + 0.192·27-s − 1.25·31-s + 0.522·33-s − 0.493·37-s − 0.640·39-s + 0.312·41-s − 0.152·43-s + 0.437·47-s + 2/7·49-s − 0.140·51-s + 1.51·53-s + 0.662·57-s + 0.256·61-s + 0.377·63-s + 1.58·67-s + 0.481·69-s − 0.237·71-s + 0.702·73-s + ⋯

Functional equation

Λ(s)=(20400s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(20400s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2040020400    =    24⋅3⋅52⋅172^{4} \cdot 3 \cdot 5^{2} \cdot 17
Sign: 11
Analytic conductor: 162.894162.894
Root analytic conductor: 12.763012.7630
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 20400, ( :1/2), 1)(2,\ 20400,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.6812822403.681282240
L(12)L(\frac12) ≈\approx 3.6812822403.681282240
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−T 1 - T
5 1 1
17 1+T 1 + T
good7 1−3T+pT2 1 - 3 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−13T+pT2 1 - 13 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{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

−15.46474212626249, −14.96297523926415, −14.49614770013870, −14.16904510509826, −13.64820767222871, −12.86616577702567, −12.38229417910952, −11.69262019055924, −11.36149405571179, −10.70779708535053, −9.964304837581642, −9.422291760353635, −8.939913669742462, −8.361331911739950, −7.688066267003204, −7.150551089805227, −6.798064529210768, −5.613866491566717, −5.201348397266471, −4.512326375835483, −3.867088847220897, −3.122263798256060, −2.283594157037941, −1.644224005745131, −0.7955385103841521, 0.7955385103841521, 1.644224005745131, 2.283594157037941, 3.122263798256060, 3.867088847220897, 4.512326375835483, 5.201348397266471, 5.613866491566717, 6.798064529210768, 7.150551089805227, 7.688066267003204, 8.361331911739950, 8.939913669742462, 9.422291760353635, 9.964304837581642, 10.70779708535053, 11.36149405571179, 11.69262019055924, 12.38229417910952, 12.86616577702567, 13.64820767222871, 14.16904510509826, 14.49614770013870, 14.96297523926415, 15.46474212626249

Graph of the ZZ-function along the critical line