Properties

 Label 8-570e4-1.1-c1e4-0-6 Degree $8$ Conductor $105560010000$ Sign $1$ Analytic cond. $429.148$ Root an. cond. $2.13341$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·4-s + 2·9-s + 3·16-s + 4·19-s + 8·25-s + 24·29-s − 4·36-s − 24·41-s + 28·49-s + 24·59-s + 40·61-s − 4·64-s − 48·71-s − 8·76-s − 5·81-s + 48·89-s − 16·100-s − 48·116-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 − 4-s + 2/3·9-s + 3/4·16-s + 0.917·19-s + 8/5·25-s + 4.45·29-s − 2/3·36-s − 3.74·41-s + 4·49-s + 3.12·59-s + 5.12·61-s − 1/2·64-s − 5.69·71-s − 0.917·76-s − 5/9·81-s + 5.08·89-s − 8/5·100-s − 4.45·116-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$429.148$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{570} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.904797835$$ $$L(\frac12)$$ $$\approx$$ $$2.904797835$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
5$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
good7$C_2$ $$( 1 - p T^{2} )^{4}$$
11$C_2$ $$( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}$$
13$C_2^2$ $$( 1 + 8 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2$ $$( 1 + p T^{2} )^{4}$$
23$C_2^2$ $$( 1 + 28 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 - 44 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 56 T^{2} + p^{2} T^{4} )^{2}$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
43$C_2^2$ $$( 1 + 58 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 68 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
61$C_2$ $$( 1 - 10 T + p T^{2} )^{4}$$
67$C_2$ $$( 1 + p T^{2} )^{4}$$
71$C_2$ $$( 1 + 12 T + p T^{2} )^{4}$$
73$C_2$ $$( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2}$$
79$C_2^2$ $$( 1 + 4 T^{2} + p^{2} T^{4} )^{2}$$
83$C_2$ $$( 1 + p T^{2} )^{4}$$
89$C_2$ $$( 1 - 12 T + p T^{2} )^{4}$$
97$C_2$ $$( 1 + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$