# Properties

 Degree 4 Conductor $2^{7} \cdot 5^{2} \cdot 19^{2}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·5-s − 6·9-s + 4·17-s + 4·19-s + 3·25-s − 16·31-s − 12·45-s + 2·49-s − 8·59-s − 4·61-s + 16·67-s − 12·73-s + 27·81-s + 8·85-s + 8·95-s + 12·101-s + 8·103-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s − 32·155-s + ⋯
 L(s)  = 1 + 0.894·5-s − 2·9-s + 0.970·17-s + 0.917·19-s + 3/5·25-s − 2.87·31-s − 1.78·45-s + 2/7·49-s − 1.04·59-s − 0.512·61-s + 1.95·67-s − 1.40·73-s + 3·81-s + 0.867·85-s + 0.820·95-s + 1.19·101-s + 0.788·103-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s − 2.57·155-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1155200$$    =    $$2^{7} \cdot 5^{2} \cdot 19^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{1155200} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = $$1$$ Selberg data = $$(4,\ 1155200,\ (\ :1/2, 1/2),\ -1)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5,\;19\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
5$C_1$ $$( 1 - T )^{2}$$
19$C_2$ $$1 - 4 T + p T^{2}$$
good3$C_2$ $$( 1 + p T^{2} )^{2}$$
7$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
29$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
31$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
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$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
\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}