# Properties

 Degree 4 Conductor $2^{7} \cdot 5^{2} \cdot 11^{2}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·5-s − 8·7-s − 6·9-s + 4·11-s + 8·19-s + 3·25-s − 16·35-s + 12·37-s − 16·43-s − 12·45-s + 34·49-s + 12·53-s + 8·55-s + 48·63-s − 32·77-s + 27·81-s − 32·83-s − 12·89-s + 16·95-s − 28·97-s − 24·99-s + 36·113-s + 5·121-s + 4·125-s + 127-s + 131-s − 64·133-s + ⋯
 L(s)  = 1 + 0.894·5-s − 3.02·7-s − 2·9-s + 1.20·11-s + 1.83·19-s + 3/5·25-s − 2.70·35-s + 1.97·37-s − 2.43·43-s − 1.78·45-s + 34/7·49-s + 1.64·53-s + 1.07·55-s + 6.04·63-s − 3.64·77-s + 3·81-s − 3.51·83-s − 1.27·89-s + 1.64·95-s − 2.84·97-s − 2.41·99-s + 3.38·113-s + 5/11·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$387200$$    =    $$2^{7} \cdot 5^{2} \cdot 11^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{387200} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 1 Selberg data = $(4,\ 387200,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;11\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 $$1$$
5$C_1$ $$( 1 - T )^{2}$$
11$C_2$ $$1 - 4 T + p T^{2}$$
good3$C_2$ $$( 1 + p T^{2} )^{2}$$
7$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
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$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
53$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 + 16 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 + 14 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}