# Properties

 Degree $4$ Conductor $12446784$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s − 6-s + 8-s + 11-s − 16-s + 2·17-s + 2·19-s − 22-s + 24-s − 25-s − 27-s + 33-s − 2·34-s − 2·38-s − 41-s + 43-s − 48-s + 50-s + 2·51-s + 54-s + 2·57-s − 59-s + 64-s − 66-s + 67-s + 2·73-s + ⋯
 L(s)  = 1 − 2-s + 3-s − 6-s + 8-s + 11-s − 16-s + 2·17-s + 2·19-s − 22-s + 24-s − 25-s − 27-s + 33-s − 2·34-s − 2·38-s − 41-s + 43-s − 48-s + 50-s + 2·51-s + 54-s + 2·57-s − 59-s + 64-s − 66-s + 67-s + 2·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$12446784$$    =    $$2^{6} \cdot 3^{4} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{3528} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 12446784,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.319634231$$ $$L(\frac12)$$ $$\approx$$ $$1.319634231$$ $$L(1)$$ 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 $$1$$
good5$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
11$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
13$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
17$C_2$ $$( 1 - T + T^{2} )^{2}$$
19$C_2$ $$( 1 - T + T^{2} )^{2}$$
23$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
29$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
31$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
37$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
41$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
43$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
47$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
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_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
73$C_2$ $$( 1 - T + T^{2} )^{2}$$
79$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
83$C_2$ $$( 1 - T + T^{2} )^{2}$$
89$C_1$ $$( 1 + T )^{4}$$
97$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$