Properties

Label 8-45e8-1.1-c0e4-0-1
Degree $8$
Conductor $1.682\times 10^{13}$
Sign $1$
Analytic cond. $1.04310$
Root an. cond. $1.00528$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 16-s + 4·19-s + 2·31-s + 2·49-s + 2·61-s − 2·64-s − 4·76-s − 2·79-s + 4·109-s − 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 4-s + 16-s + 4·19-s + 2·31-s + 2·49-s + 2·61-s − 2·64-s − 4·76-s − 2·79-s + 4·109-s − 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.04310\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.375176662\)
\(L(\frac12)\) \(\approx\) \(1.375176662\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
83$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.88104858651300994514661298742, −6.41247542612574389861234128499, −6.07955915592130985422612419225, −6.06028525691658715527193952385, −5.91177179035889751102159309482, −5.53790045439090630797049024542, −5.41546441004578799070022287827, −5.22382278892486172834765675987, −5.12979085574737022115734054580, −4.76106910382948079635209442445, −4.45715624614404614809135921284, −4.38599537733489761626788181243, −4.37285045112079370579871923867, −3.61861895121589991387926485869, −3.57076606916853206557710788218, −3.54517670398687785612810433522, −3.28404949262021698157272904817, −2.78742674956118604913233033635, −2.78490642555186432135423112300, −2.42929120577766331274259350851, −2.15575717556613149128907223841, −1.59317375478164994171015730582, −1.13816103219755869098658394721, −1.10275190080177178849316301460, −0.796436407443435296753350459483, 0.796436407443435296753350459483, 1.10275190080177178849316301460, 1.13816103219755869098658394721, 1.59317375478164994171015730582, 2.15575717556613149128907223841, 2.42929120577766331274259350851, 2.78490642555186432135423112300, 2.78742674956118604913233033635, 3.28404949262021698157272904817, 3.54517670398687785612810433522, 3.57076606916853206557710788218, 3.61861895121589991387926485869, 4.37285045112079370579871923867, 4.38599537733489761626788181243, 4.45715624614404614809135921284, 4.76106910382948079635209442445, 5.12979085574737022115734054580, 5.22382278892486172834765675987, 5.41546441004578799070022287827, 5.53790045439090630797049024542, 5.91177179035889751102159309482, 6.06028525691658715527193952385, 6.07955915592130985422612419225, 6.41247542612574389861234128499, 6.88104858651300994514661298742

Graph of the $Z$-function along the critical line