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Properties

Label 4-160e2-1.1-c1e2-0-18
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 − 2·9-s − 12·13-s + 4·17-s + 3·25-s − 4·29-s + 4·37-s − 20·41-s + 4·45-s − 10·49-s + 4·53-s + 4·61-s + 24·65-s + 20·73-s − 5·81-s − 8·85-s − 12·89-s + 20·97-s + 28·101-s − 28·109-s − 12·113-s + 24·117-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 2/3·9-s − 3.32·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s + 0.657·37-s − 3.12·41-s + 0.596·45-s − 1.42·49-s + 0.549·53-s + 0.512·61-s + 2.97·65-s + 2.34·73-s − 5/9·81-s − 0.867·85-s − 1.27·89-s + 2.03·97-s + 2.78·101-s − 2.68·109-s − 1.12·113-s + 2.21·117-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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
5C1C_1 (1+T)2 ( 1 + T )^{2}
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
43C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
47C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
53C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−10T+pT2)2 ( 1 - 10 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

−10.29364010457156446481778884088, −9.759510932768572667567645779831, −9.572587619444812153159504270402, −8.731345173958242569557384315653, −7.967338729692305876626971946668, −7.85140754717246485615619273252, −7.05269675268361745315034445294, −6.81951269321906735272373937416, −5.70243463207358237884993422463, −4.92945525258561389220988501216, −4.88847555485350104575427844162, −3.69518610983128083383746206280, −3.01064345970677263239408332098, −2.17341817164904427445574627486, 0, 2.17341817164904427445574627486, 3.01064345970677263239408332098, 3.69518610983128083383746206280, 4.88847555485350104575427844162, 4.92945525258561389220988501216, 5.70243463207358237884993422463, 6.81951269321906735272373937416, 7.05269675268361745315034445294, 7.85140754717246485615619273252, 7.967338729692305876626971946668, 8.731345173958242569557384315653, 9.572587619444812153159504270402, 9.759510932768572667567645779831, 10.29364010457156446481778884088

Graph of the ZZ-function along the critical line