# Properties

 Label 4-168e2-1.1-c0e2-0-1 Degree $4$ Conductor $28224$ Sign $1$ Analytic cond. $0.00702963$ Root an. cond. $0.289556$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 11-s − 14-s − 15-s − 16-s − 21-s − 22-s − 24-s + 25-s − 27-s + 2·29-s − 30-s + 31-s − 33-s + 35-s + 40-s − 42-s − 48-s + 50-s − 53-s − 54-s + ⋯
 L(s)  = 1 + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 11-s − 14-s − 15-s − 16-s − 21-s − 22-s − 24-s + 25-s − 27-s + 2·29-s − 30-s + 31-s − 33-s + 35-s + 40-s − 42-s − 48-s + 50-s − 53-s − 54-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$28224$$    =    $$2^{6} \cdot 3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$0.00702963$$ Root analytic conductor: $$0.289556$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{168} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 28224,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6320305171$$ $$L(\frac12)$$ $$\approx$$ $$0.6320305171$$ $$L(1)$$ 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$C_2$ $$1 - T + T^{2}$$
3$C_2$ $$1 - T + T^{2}$$
7$C_2$ $$1 + T + T^{2}$$
good5$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
11$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
13$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
17$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
19$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
23$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
29$C_2$ $$( 1 - T + T^{2} )^{2}$$
31$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
37$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
41$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
47$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
53$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
59$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
61$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
67$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
73$C_2$ $$( 1 + T + T^{2} )^{2}$$
79$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
83$C_2$ $$( 1 - T + T^{2} )^{2}$$
89$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
97$C_2$ $$( 1 + T + T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$