# Properties

 Label 8-507e4-1.1-c1e4-0-11 Degree $8$ Conductor $66074188401$ Sign $1$ Analytic cond. $268.621$ Root an. cond. $2.01206$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·7-s − 3·9-s + 4·16-s + 2·19-s − 14·31-s + 22·37-s + 6·43-s + 23·49-s − 24·63-s − 10·67-s − 34·73-s + 28·97-s + 38·109-s + 32·112-s + 127-s + 131-s + 16·133-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·171-s + 173-s + ⋯
 L(s)  = 1 + 3.02·7-s − 9-s + 16-s + 0.458·19-s − 2.51·31-s + 3.61·37-s + 0.914·43-s + 23/7·49-s − 3.02·63-s − 1.22·67-s − 3.97·73-s + 2.84·97-s + 3.63·109-s + 3.02·112-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.458·171-s + 0.0760·173-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$268.621$$ Root analytic conductor: $$2.01206$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{507} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.659169431$$ $$L(\frac12)$$ $$\approx$$ $$3.659169431$$ $$L(\frac{3}{2})$$ 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)$
bad3$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
13 $$1$$
good2$C_2^2$$\times$$C_2^2$ $$( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )$$
5$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
7$C_2$$\times$$C_2^2$ $$( 1 - 4 T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} )$$
11$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
17$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
19$C_2$$\times$$C_2^2$ $$( 1 - T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} )$$
23$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
31$C_2$$\times$$C_2^2$ $$( 1 + 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} )$$
37$C_2$$\times$$C_2^2$ $$( 1 - 11 T + p T^{2} )^{2}( 1 - 73 T^{2} + p^{2} T^{4} )$$
41$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2}$$
47$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
53$C_2$ $$( 1 - p T^{2} )^{4}$$
59$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
61$C_2^2$$\times$$C_2^2$ $$( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} )$$
67$C_2$$\times$$C_2^2$ $$( 1 + 5 T + p T^{2} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} )$$
71$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
73$C_2$$\times$$C_2^2$ $$( 1 + 17 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} )$$
79$C_2^2$ $$( 1 + 11 T^{2} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
89$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
97$C_2$$\times$$C_2^2$ $$( 1 - 14 T + p T^{2} )^{2}( 1 + 167 T^{2} + p^{2} T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$