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Properties

Label 4-160e2-1.1-c1e2-0-17
Degree 44
Conductor 2560025600
Sign −1-1
Analytic cond. 1.632271.63227
Root an. cond. 1.130311.13031
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 6·9-s − 25-s − 20·29-s + 20·41-s + 12·45-s − 14·49-s − 20·61-s + 27·81-s + 20·89-s − 4·101-s + 12·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 0.894·5-s − 2·9-s − 1/5·25-s − 3.71·29-s + 3.12·41-s + 1.78·45-s − 2·49-s − 2.56·61-s + 3·81-s + 2.11·89-s − 0.398·101-s + 1.14·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

Λ(s)=(25600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(25600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2560025600    =    210⋅522^{10} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 1.632271.63227
Root analytic conductor: 1.130311.13031
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 25600, ( :1/2,1/2), −1)(4,\ 25600,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{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

−10.84453522425202397161354104633, −9.617316516935557446365175892726, −9.206579341791145473157585005601, −8.955386231165229198073332132052, −8.025782606763976976877695514876, −7.77199473906097062385997282225, −7.38383189330491254934984805094, −6.27997548582365804353754428543, −5.87146418848833687506982135026, −5.37327377173791634385808352079, −4.46989807490289628768060263585, −3.67478222653086463350186782835, −3.13241440937412885242101141742, −2.12458086750050120053975349819, 0, 2.12458086750050120053975349819, 3.13241440937412885242101141742, 3.67478222653086463350186782835, 4.46989807490289628768060263585, 5.37327377173791634385808352079, 5.87146418848833687506982135026, 6.27997548582365804353754428543, 7.38383189330491254934984805094, 7.77199473906097062385997282225, 8.025782606763976976877695514876, 8.955386231165229198073332132052, 9.206579341791145473157585005601, 9.617316516935557446365175892726, 10.84453522425202397161354104633

Graph of the ZZ-function along the critical line