# Properties

 Degree $4$ Conductor $451584$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3-s − 5·7-s + 2·11-s + 2·13-s + 2·17-s − 5·19-s − 5·21-s − 6·23-s + 5·25-s − 27-s − 16·29-s + 3·31-s + 2·33-s + 9·37-s + 2·39-s + 4·41-s + 2·43-s − 8·47-s + 18·49-s + 2·51-s − 6·53-s − 5·57-s − 6·59-s + 2·61-s + 5·67-s − 6·69-s + 8·71-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.88·7-s + 0.603·11-s + 0.554·13-s + 0.485·17-s − 1.14·19-s − 1.09·21-s − 1.25·23-s + 25-s − 0.192·27-s − 2.97·29-s + 0.538·31-s + 0.348·33-s + 1.47·37-s + 0.320·39-s + 0.624·41-s + 0.304·43-s − 1.16·47-s + 18/7·49-s + 0.280·51-s − 0.824·53-s − 0.662·57-s − 0.781·59-s + 0.256·61-s + 0.610·67-s − 0.722·69-s + 0.949·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$451584$$    =    $$2^{10} \cdot 3^{2} \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{672} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 451584,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.44558$$ $$L(\frac12)$$ $$\approx$$ $$1.44558$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_2$ $$1 - T + T^{2}$$
7$C_2$ $$1 + 5 T + p T^{2}$$
good5$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 - T + p T^{2} )^{2}$$
17$C_2^2$ $$1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
23$C_2^2$ $$1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
29$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
31$C_2^2$ $$1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
37$C_2^2$ $$1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
41$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - T + p T^{2} )^{2}$$
47$C_2^2$ $$1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
71$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
73$C_2^2$ $$1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
79$C_2^2$ $$1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2^2$ $$1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 18 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$