# Properties

 Degree 4 Conductor $2^{4} \cdot 3^{6}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 4·5-s + 6·7-s − 8·8-s + 8·10-s − 21·11-s + 22·13-s + 12·14-s − 16·16-s + 22·17-s − 42·22-s + 42·23-s + 25·25-s + 44·26-s + 34·29-s − 12·31-s + 44·34-s + 24·35-s − 32·37-s − 32·40-s + 13·41-s + 87·43-s + 84·46-s + 6·47-s − 25·49-s + 50·50-s − 104·53-s + ⋯
 L(s)  = 1 + 2-s + 4/5·5-s + 6/7·7-s − 8-s + 4/5·10-s − 1.90·11-s + 1.69·13-s + 6/7·14-s − 16-s + 1.29·17-s − 1.90·22-s + 1.82·23-s + 25-s + 1.69·26-s + 1.17·29-s − 0.387·31-s + 1.29·34-s + 0.685·35-s − 0.864·37-s − 4/5·40-s + 0.317·41-s + 2.02·43-s + 1.82·46-s + 6/47·47-s − 0.510·49-s + 50-s − 1.96·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$11664$$    =    $$2^{4} \cdot 3^{6}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{108} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 11664,\ (\ :1, 1),\ 1)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.93085$$ $$L(\frac12)$$ $$\approx$$ $$2.93085$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 - p T + p^{2} T^{2}$$
3 $$1$$
good5$C_2^2$ $$1 - 4 T - 9 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4}$$
7$C_2^2$ $$1 - 6 T + 61 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4}$$
11$C_2^2$ $$1 + 21 T + 268 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4}$$
13$C_2$ $$( 1 - 23 T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} )$$
17$C_2$ $$( 1 - 11 T + p^{2} T^{2} )^{2}$$
19$C_2^2$ $$1 - 479 T^{2} + p^{4} T^{4}$$
23$C_2^2$ $$1 - 42 T + 1117 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4}$$
29$C_2^2$ $$1 - 34 T + 315 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4}$$
31$C_2^2$ $$1 + 12 T + 1009 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4}$$
37$C_2$ $$( 1 + 16 T + p^{2} T^{2} )^{2}$$
41$C_2^2$ $$1 - 13 T - 1512 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4}$$
43$C_2^2$ $$1 - 87 T + 4372 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4}$$
47$C_2^2$ $$1 - 6 T + 2221 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4}$$
53$C_2$ $$( 1 + 52 T + p^{2} T^{2} )^{2}$$
59$C_2^2$ $$1 + 93 T + 6364 T^{2} + 93 p^{2} T^{3} + p^{4} T^{4}$$
61$C_2^2$ $$1 - 16 T - 3465 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4}$$
67$C_1$$\times$$C_2$ $$( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} )$$
71$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
73$C_2$ $$( 1 + 25 T + p^{2} T^{2} )^{2}$$
79$C_2^2$ $$1 + 48 T + 7009 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4}$$
83$C_2^2$ $$1 - 60 T + 8089 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4}$$
89$C_2$ $$( 1 - 2 T + p^{2} T^{2} )^{2}$$
97$C_2^2$ $$1 - 43 T - 7560 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}