# Properties

 Degree $4$ Conductor $42336$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 8·11-s + 12-s + 12·13-s + 16-s − 18-s − 8·22-s − 16·23-s − 24-s − 6·25-s − 12·26-s + 27-s − 32-s + 8·33-s + 36-s − 20·37-s + 12·39-s + 8·44-s + 16·46-s + 48-s + 49-s + 6·50-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 2.41·11-s + 0.288·12-s + 3.32·13-s + 1/4·16-s − 0.235·18-s − 1.70·22-s − 3.33·23-s − 0.204·24-s − 6/5·25-s − 2.35·26-s + 0.192·27-s − 0.176·32-s + 1.39·33-s + 1/6·36-s − 3.28·37-s + 1.92·39-s + 1.20·44-s + 2.35·46-s + 0.144·48-s + 1/7·49-s + 0.848·50-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 42336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 42336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$42336$$    =    $$2^{5} \cdot 3^{3} \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{42336} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 42336,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.435805038$$ $$L(\frac12)$$ $$\approx$$ $$1.435805038$$ $$L(\frac{3}{2})$$ 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$C_1$ $$1 + T$$
3$C_1$ $$1 - T$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
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$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 + 14 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}$$