# Properties

 Label 4-6e6-1.1-c1e2-0-17 Degree $4$ Conductor $46656$ Sign $-1$ Analytic cond. $2.97482$ Root an. cond. $1.31330$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 3·7-s − 2·13-s − 12·19-s − 25-s − 8·37-s − 49-s + 4·61-s + 6·67-s + 73-s + 6·79-s + 6·91-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 5·121-s + 127-s + 131-s + 36·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
 L(s)  = 1 − 1.13·7-s − 0.554·13-s − 2.75·19-s − 1/5·25-s − 1.31·37-s − 1/7·49-s + 0.512·61-s + 0.733·67-s + 0.117·73-s + 0.675·79-s + 0.628·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$46656$$    =    $$2^{6} \cdot 3^{6}$$ Sign: $-1$ Analytic conductor: $$2.97482$$ Root analytic conductor: $$1.31330$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{46656} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 46656,\ (\ :1/2, 1/2),\ -1)$$

## 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 $$1$$
3 $$1$$
good5$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
7$C_2$ $$( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2^2$ $$1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2^2$ $$1 - 35 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
59$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
71$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
73$C_2^2$ $$1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4}$$
79$C_2^2$ $$1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
83$C_2^2$ $$1 + 139 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$