# Properties

 Degree $4$ Conductor $213444$ Sign $-1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $1$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 9-s − 6·12-s + 5·16-s + 12·17-s − 2·18-s + 8·24-s − 10·25-s + 4·27-s − 12·29-s − 8·31-s − 6·32-s − 24·34-s + 3·36-s + 4·37-s + 12·41-s − 10·48-s + 49-s + 20·50-s − 24·51-s − 8·54-s + 24·58-s + 16·62-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 1/3·9-s − 1.73·12-s + 5/4·16-s + 2.91·17-s − 0.471·18-s + 1.63·24-s − 2·25-s + 0.769·27-s − 2.22·29-s − 1.43·31-s − 1.06·32-s − 4.11·34-s + 1/2·36-s + 0.657·37-s + 1.87·41-s − 1.44·48-s + 1/7·49-s + 2.82·50-s − 3.36·51-s − 1.08·54-s + 3.15·58-s + 2.03·62-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$213444$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{213444} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 213444,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad2$C_1$ $$( 1 + T )^{2}$$
3$C_2$ $$1 + 2 T + p T^{2}$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
11$C_2$ $$1 + p T^{2}$$
good5$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
61$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
79$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}$$