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Properties

Label 2-20400-1.1-c1-0-18
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·7-s + 9-s − 3·11-s − 17-s + 7·19-s − 3·21-s + 4·23-s + 27-s + 29-s + 10·31-s − 3·33-s + 37-s − 3·41-s − 10·43-s + 3·47-s + 2·49-s − 51-s − 9·53-s + 7·57-s + 10·59-s − 12·61-s − 3·63-s + 10·67-s + 4·69-s − 13·73-s + 9·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.242·17-s + 1.60·19-s − 0.654·21-s + 0.834·23-s + 0.192·27-s + 0.185·29-s + 1.79·31-s − 0.522·33-s + 0.164·37-s − 0.468·41-s − 1.52·43-s + 0.437·47-s + 2/7·49-s − 0.140·51-s − 1.23·53-s + 0.927·57-s + 1.30·59-s − 1.53·61-s − 0.377·63-s + 1.22·67-s + 0.481·69-s − 1.52·73-s + 1.02·77-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 2.0557936142.055793614
L(12)L(\frac12) ≈\approx 2.0557936142.055793614
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+pT2 1 + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−T+pT2 1 - T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+13T+pT2 1 + 13 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−6T+pT2 1 - 6 T + p T^{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

−15.58853527319409, −15.29924895239724, −14.53116815933340, −13.82101503462610, −13.49315274718983, −13.04258795468094, −12.47933952475535, −11.83168459995481, −11.29149683579348, −10.45080059638628, −9.960064256143718, −9.627318101015784, −8.954176622252083, −8.298835197148789, −7.783889913377972, −7.068190577649401, −6.624585864206645, −5.889922366753494, −5.119541355270877, −4.604903418455253, −3.606705512857884, −2.982270131464028, −2.732056956770397, −1.554151940525186, −0.5742687454176715, 0.5742687454176715, 1.554151940525186, 2.732056956770397, 2.982270131464028, 3.606705512857884, 4.604903418455253, 5.119541355270877, 5.889922366753494, 6.624585864206645, 7.068190577649401, 7.783889913377972, 8.298835197148789, 8.954176622252083, 9.627318101015784, 9.960064256143718, 10.45080059638628, 11.29149683579348, 11.83168459995481, 12.47933952475535, 13.04258795468094, 13.49315274718983, 13.82101503462610, 14.53116815933340, 15.29924895239724, 15.58853527319409

Graph of the ZZ-function along the critical line