# Properties

 Degree $4$ Conductor $2480625$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s − 7-s + 3·19-s − 28-s + 37-s + 4·43-s − 3·61-s − 64-s − 67-s − 3·73-s + 3·76-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯
 L(s)  = 1 + 4-s − 7-s + 3·19-s − 28-s + 37-s + 4·43-s − 3·61-s − 64-s − 67-s − 3·73-s + 3·76-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2480625$$    =    $$3^{4} \cdot 5^{4} \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{1575} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 2480625,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.410286689$$ $$L(\frac12)$$ $$\approx$$ $$1.410286689$$ $$L(1)$$ 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)$
bad3 $$1$$
5 $$1$$
7$C_2$ $$1 + T + T^{2}$$
good2$C_2^2$ $$1 - T^{2} + T^{4}$$
11$C_2^2$ $$1 - T^{2} + T^{4}$$
13$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
17$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
19$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 - T + T^{2} )$$
23$C_2^2$ $$1 - T^{2} + T^{4}$$
29$C_2$ $$( 1 + T^{2} )^{2}$$
31$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
37$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
41$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
43$C_1$ $$( 1 - T )^{4}$$
47$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
53$C_2^2$ $$1 - T^{2} + T^{4}$$
59$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
61$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T + T^{2} )$$
67$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
71$C_2$ $$( 1 + T^{2} )^{2}$$
73$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T + T^{2} )$$
79$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
83$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
89$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
97$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$