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Properties

Label 4-3528e2-1.1-c1e2-0-3
Degree 44
Conductor 1244678412446784
Sign 11
Analytic cond. 793.617793.617
Root an. cond. 5.307655.30765
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·11-s − 4·13-s − 6·17-s − 4·19-s + 6·23-s + 5·25-s − 4·31-s − 10·37-s + 4·41-s − 8·43-s − 4·47-s − 12·53-s − 4·55-s − 12·59-s + 6·61-s + 8·65-s + 4·67-s + 28·71-s − 2·73-s + 8·79-s − 32·83-s + 12·85-s + 6·89-s + 8·95-s + 36·97-s − 14·101-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.603·11-s − 1.10·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s + 25-s − 0.718·31-s − 1.64·37-s + 0.624·41-s − 1.21·43-s − 0.583·47-s − 1.64·53-s − 0.539·55-s − 1.56·59-s + 0.768·61-s + 0.992·65-s + 0.488·67-s + 3.32·71-s − 0.234·73-s + 0.900·79-s − 3.51·83-s + 1.30·85-s + 0.635·89-s + 0.820·95-s + 3.65·97-s − 1.39·101-s + ⋯

Functional equation

Λ(s)=(12446784s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(12446784s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1244678412446784    =    26⋅34⋅742^{6} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 793.617793.617
Root analytic conductor: 5.307655.30765
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 12446784, ( :1/2,1/2), 1)(4,\ 12446784,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.30906957490.3090695749
L(12)L(\frac12) ≈\approx 0.30906957490.3090695749
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
7 1 1
good5C22C_2^2 1+2T−T2+2pT3+p2T4 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−2T−7T2−2pT3+p2T4 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C22C_2^2 1+6T+19T2+6pT3+p2T4 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+4T−3T2+4pT3+p2T4 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−6T+13T2−6pT3+p2T4 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−7T+pT2)(1+11T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} )
37C2C_2 (1−T+pT2)(1+11T+pT2) ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} )
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1+4T−31T2+4pT3+p2T4 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+12T+91T2+12pT3+p2T4 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+12T+85T2+12pT3+p2T4 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−6T−25T2−6pT3+p2T4 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−4T−51T2−4pT3+p2T4 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
73C22C_2^2 1+2T−69T2+2pT3+p2T4 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−8T−15T2−8pT3+p2T4 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
89C22C_2^2 1−6T−53T2−6pT3+p2T4 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4}
97C2C_2 (1−18T+pT2)2 ( 1 - 18 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.892532111206215314081814045533, −8.259823584698242129600574003328, −8.158978306695114281518236043958, −7.55504578972573212675448590662, −7.33250340666059212817064818339, −6.75487803874930213311765613616, −6.61089258524233622314130760797, −6.49051977246742497052624035502, −5.72059138298391011076206441038, −5.12040408277451557448104379783, −4.93856985383457531089945309857, −4.59953846223430251767407471453, −4.19561667137027706637369340172, −3.56393132338580887204049919605, −3.43227728945779337899918337910, −2.78608199452191970089853165671, −2.24915439229303985777566669068, −1.83716675895772120153222614565, −1.13150254014342935058229240718, −0.17587506248643707713497789691, 0.17587506248643707713497789691, 1.13150254014342935058229240718, 1.83716675895772120153222614565, 2.24915439229303985777566669068, 2.78608199452191970089853165671, 3.43227728945779337899918337910, 3.56393132338580887204049919605, 4.19561667137027706637369340172, 4.59953846223430251767407471453, 4.93856985383457531089945309857, 5.12040408277451557448104379783, 5.72059138298391011076206441038, 6.49051977246742497052624035502, 6.61089258524233622314130760797, 6.75487803874930213311765613616, 7.33250340666059212817064818339, 7.55504578972573212675448590662, 8.158978306695114281518236043958, 8.259823584698242129600574003328, 8.892532111206215314081814045533

Graph of the ZZ-function along the critical line