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Properties

Label 2-40-40.19-c10-0-31
Degree 22
Conductor 4040
Sign 11
Analytic cond. 25.414225.4142
Root an. cond. 5.041255.04125
Motivic weight 1010
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s + 1.02e3·4-s − 3.12e3·5-s − 2.68e4·7-s + 3.27e4·8-s + 5.90e4·9-s − 1.00e5·10-s + 3.21e5·11-s + 6.82e5·13-s − 8.60e5·14-s + 1.04e6·16-s + 1.88e6·18-s − 1.28e6·19-s − 3.20e6·20-s + 1.02e7·22-s + 1.77e6·23-s + 9.76e6·25-s + 2.18e7·26-s − 2.75e7·28-s + 3.35e7·32-s + 8.40e7·35-s + 6.04e7·36-s − 1.12e8·37-s − 4.12e7·38-s − 1.02e8·40-s − 6.11e5·41-s + 3.28e8·44-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s − 1.59·7-s + 8-s + 9-s − 10-s + 1.99·11-s + 1.83·13-s − 1.59·14-s + 16-s + 18-s − 0.520·19-s − 20-s + 1.99·22-s + 0.275·23-s + 25-s + 1.83·26-s − 1.59·28-s + 32-s + 1.59·35-s + 36-s − 1.62·37-s − 0.520·38-s − 40-s − 0.00527·41-s + 1.99·44-s + ⋯

Functional equation

Λ(s)=(40s/2 ΓC(s) L(s)=( Λ(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}
Λ(s)=(40s/2 ΓC(s+5) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4040    =    23⋅52^{3} \cdot 5
Sign: 11
Analytic conductor: 25.414225.4142
Root analytic conductor: 5.041255.04125
Motivic weight: 1010
Rational: yes
Arithmetic: yes
Character: χ40(19,⋅)\chi_{40} (19, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 40, ( :5), 1)(2,\ 40,\ (\ :5),\ 1)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 3.4843968283.484396828
L(12)L(\frac12) ≈\approx 3.4843968283.484396828
L(6)L(6) 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−p5T 1 - p^{5} T
5 1+p5T 1 + p^{5} T
good3 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
7 1+26886T+p10T2 1 + 26886 T + p^{10} T^{2}
11 1−321102T+p10T2 1 - 321102 T + p^{10} T^{2}
13 1−682086T+p10T2 1 - 682086 T + p^{10} T^{2}
17 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
19 1+1288802T+p10T2 1 + 1288802 T + p^{10} T^{2}
23 1−1772186T+p10T2 1 - 1772186 T + p^{10} T^{2}
29 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
31 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
37 1+112652586T+p10T2 1 + 112652586 T + p^{10} T^{2}
41 1+611598T+p10T2 1 + 611598 T + p^{10} T^{2}
43 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
47 1−222705514T+p10T2 1 - 222705514 T + p^{10} T^{2}
53 1−552886486T+p10T2 1 - 552886486 T + p^{10} T^{2}
59 1+646632402T+p10T2 1 + 646632402 T + p^{10} T^{2}
61 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
67 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
71 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
73 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
79 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
83 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
89 1−5417714898T+p10T2 1 - 5417714898 T + p^{10} T^{2}
97 (1−p5T)(1+p5T) ( 1 - p^{5} T )( 1 + p^{5} T )
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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

−13.76553475858875280288932855035, −12.73946846592560551657269579053, −11.86897117581987799054642231443, −10.58660702724215720709017805445, −8.945206105222637012804815076152, −6.97087349537319414664087226827, −6.31427518913810964724375156592, −4.00679183679950424192685361831, −3.55651034573834383959461642706, −1.17161597705916422204989183659, 1.17161597705916422204989183659, 3.55651034573834383959461642706, 4.00679183679950424192685361831, 6.31427518913810964724375156592, 6.97087349537319414664087226827, 8.945206105222637012804815076152, 10.58660702724215720709017805445, 11.86897117581987799054642231443, 12.73946846592560551657269579053, 13.76553475858875280288932855035

Graph of the ZZ-function along the critical line