# Properties

 Label 4-826875-1.1-c1e2-0-19 Degree $4$ Conductor $826875$ Sign $-1$ Analytic cond. $52.7222$ Root an. cond. $2.69462$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3-s − 3·4-s + 9-s − 3·12-s + 5·16-s − 4·17-s + 27-s − 3·36-s + 20·37-s + 20·41-s − 8·43-s − 16·47-s + 5·48-s − 7·49-s − 4·51-s − 8·59-s − 3·64-s − 24·67-s + 12·68-s + 81-s − 24·83-s − 12·89-s + 12·101-s − 3·108-s + 28·109-s + 20·111-s − 6·121-s + ⋯
 L(s)  = 1 + 0.577·3-s − 3/2·4-s + 1/3·9-s − 0.866·12-s + 5/4·16-s − 0.970·17-s + 0.192·27-s − 1/2·36-s + 3.28·37-s + 3.12·41-s − 1.21·43-s − 2.33·47-s + 0.721·48-s − 49-s − 0.560·51-s − 1.04·59-s − 3/8·64-s − 2.93·67-s + 1.45·68-s + 1/9·81-s − 2.63·83-s − 1.27·89-s + 1.19·101-s − 0.288·108-s + 2.68·109-s + 1.89·111-s − 0.545·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$826875$$    =    $$3^{3} \cdot 5^{4} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$52.7222$$ Root analytic conductor: $$2.69462$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{826875} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 826875,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$C_1$ $$1 - T$$
5 $$1$$
7$C_2$ $$1 + p T^{2}$$
good2$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
59$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$