# Properties

 Label 8-273e4-1.1-c1e4-0-8 Degree $8$ Conductor $5554571841$ Sign $1$ Analytic cond. $22.5818$ Root an. cond. $1.47645$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·9-s + 4·16-s + 2·19-s − 8·31-s − 20·37-s + 36·43-s − 11·49-s − 22·67-s + 14·73-s + 27·81-s + 10·97-s − 12·103-s + 38·109-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 12·171-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 2·9-s + 16-s + 0.458·19-s − 1.43·31-s − 3.28·37-s + 5.48·43-s − 1.57·49-s − 2.68·67-s + 1.63·73-s + 3·81-s + 1.01·97-s − 1.18·103-s + 3.63·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.917·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\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 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 7^{4} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$22.5818$$ Root analytic conductor: $$1.47645$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{273} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.351247120$$ $$L(\frac12)$$ $$\approx$$ $$2.351247120$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - p T^{2} )^{2}$$
7$C_2^2$ $$1 + 11 T^{2} + p^{2} T^{4}$$
13$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
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$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
19$C_2$$\times$$C_2^2$ $$( 1 - T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} )$$
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$ $$( 1 + 4 T + p T^{2} )^{2}( 1 + 59 T^{2} + p^{2} T^{4} )$$
37$C_2$$\times$$C_2^2$ $$( 1 + 10 T + p T^{2} )^{2}( 1 - 73 T^{2} + p^{2} T^{4} )$$
41$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
43$C_2$ $$( 1 - 13 T + p T^{2} )^{2}( 1 - 5 T + p T^{2} )^{2}$$
47$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 - 121 T^{2} + p^{2} T^{4} )^{2}$$
67$C_2$$\times$$C_2^2$ $$( 1 + 11 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} )$$
71$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
73$C_2$$\times$$C_2^2$ $$( 1 - 7 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} )$$
79$C_2^2$$\times$$C_2^2$ $$( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} )$$
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$ $$( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$