Properties

 Label 8-2160e4-1.1-c0e4-0-3 Degree $8$ Conductor $2.177\times 10^{13}$ Sign $1$ Analytic cond. $1.35034$ Root an. cond. $1.03825$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·4-s + 3·16-s − 4·17-s + 4·19-s + 4·31-s + 4·47-s − 4·53-s − 4·64-s + 8·68-s − 8·76-s + 4·79-s − 4·109-s + 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 8·188-s + ⋯
 L(s)  = 1 − 2·4-s + 3·16-s − 4·17-s + 4·19-s + 4·31-s + 4·47-s − 4·53-s − 4·64-s + 8·68-s − 8·76-s + 4·79-s − 4·109-s + 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 8·188-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{12} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$1.35034$$ Root analytic conductor: $$1.03825$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7941612696$$ $$L(\frac12)$$ $$\approx$$ $$0.7941612696$$ $$L(1)$$ 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)$
bad2$C_2$ $$( 1 + T^{2} )^{2}$$
3 $$1$$
5$C_2^2$ $$1 + T^{4}$$
good7$C_2^2$ $$( 1 + T^{4} )^{2}$$
11$C_2^3$ $$1 - T^{4} + T^{8}$$
13$C_2^3$ $$1 - T^{4} + T^{8}$$
17$C_2$ $$( 1 + T + T^{2} )^{4}$$
19$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T^{2} )^{2}$$
23$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
29$C_2^3$ $$1 - T^{4} + T^{8}$$
31$C_2$ $$( 1 - T + T^{2} )^{4}$$
37$C_2^2$ $$( 1 + T^{4} )^{2}$$
41$C_2$ $$( 1 + T^{2} )^{4}$$
43$C_2^3$ $$1 - T^{4} + T^{8}$$
47$C_2$ $$( 1 - T + T^{2} )^{4}$$
53$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 + T^{2} )^{2}$$
59$C_2^2$ $$( 1 + T^{4} )^{2}$$
61$C_2^2$ $$( 1 + T^{4} )^{2}$$
67$C_2^2$ $$( 1 + T^{4} )^{2}$$
71$C_2^2$ $$( 1 + T^{4} )^{2}$$
73$C_2$ $$( 1 + T^{2} )^{4}$$
79$C_2$ $$( 1 - T + T^{2} )^{4}$$
83$C_2^2$ $$( 1 + T^{4} )^{2}$$
89$C_2^2$ $$( 1 + T^{4} )^{2}$$
97$C_2^2$ $$( 1 + T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$