# Properties

 Degree 4 Conductor $3^{2} \cdot 5^{2} \cdot 19^{2}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·4-s − 2·5-s + 9-s + 5·16-s − 2·19-s + 6·20-s − 25-s + 4·29-s + 16·31-s − 3·36-s − 4·41-s − 2·45-s − 14·49-s − 24·59-s − 4·61-s − 3·64-s + 6·76-s − 10·80-s + 81-s − 4·89-s + 4·95-s + 3·100-s − 20·101-s − 20·109-s − 12·116-s − 22·121-s − 48·124-s + ⋯
 L(s)  = 1 − 3/2·4-s − 0.894·5-s + 1/3·9-s + 5/4·16-s − 0.458·19-s + 1.34·20-s − 1/5·25-s + 0.742·29-s + 2.87·31-s − 1/2·36-s − 0.624·41-s − 0.298·45-s − 2·49-s − 3.12·59-s − 0.512·61-s − 3/8·64-s + 0.688·76-s − 1.11·80-s + 1/9·81-s − 0.423·89-s + 0.410·95-s + 3/10·100-s − 1.99·101-s − 1.91·109-s − 1.11·116-s − 2·121-s − 4.31·124-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$81225$$    =    $$3^{2} \cdot 5^{2} \cdot 19^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{81225} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 1 Selberg data = $(4,\ 81225,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5,\;19\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
5$C_2$ $$1 + 2 T + p T^{2}$$
19$C_1$ $$( 1 + T )^{2}$$
good2$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
29$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
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$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
89$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
\begin{aligned} L(s) = \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} \end{aligned}