# Properties

 Degree 4 Conductor $2^{6} \cdot 3^{2} \cdot 5^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·5-s + 9-s + 8·11-s − 8·19-s − 25-s + 12·29-s + 16·31-s − 12·41-s − 2·45-s − 14·49-s − 16·55-s + 8·59-s − 4·61-s + 16·71-s − 16·79-s + 81-s − 12·89-s + 16·95-s + 8·99-s − 36·101-s − 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
 L(s)  = 1 − 0.894·5-s + 1/3·9-s + 2.41·11-s − 1.83·19-s − 1/5·25-s + 2.22·29-s + 2.87·31-s − 1.87·41-s − 0.298·45-s − 2·49-s − 2.15·55-s + 1.04·59-s − 0.512·61-s + 1.89·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 1.64·95-s + 0.804·99-s − 3.58·101-s − 0.383·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$14400$$    =    $$2^{6} \cdot 3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{14400} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 14400,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $1.039896770$ $L(\frac12)$ $\approx$ $1.039896770$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
5$C_2$ $$1 + 2 T + p T^{2}$$
good7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
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$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
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$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
59$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
\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}