# Properties

 Degree 4 Conductor $2^{12} \cdot 3^{4} \cdot 5^{2}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s + 4·41-s + 2·53-s − 4·65-s − 2·73-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
 L(s)  = 1 + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s + 4·41-s + 2·53-s − 4·65-s − 2·73-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$8294400$$    =    $$2^{12} \cdot 3^{4} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{2880} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 8294400,\ (\ :0, 0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $2.113065027$ $L(\frac12)$ $\approx$ $2.113065027$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 4. 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 $$1$$
5$C_1$ $$( 1 - T )^{2}$$
good7$C_2^2$ $$1 + T^{4}$$
11$C_2$ $$( 1 + T^{2} )^{2}$$
13$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
17$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
23$C_2^2$ $$1 + T^{4}$$
29$C_2$ $$( 1 + T^{2} )^{2}$$
31$C_2$ $$( 1 + T^{2} )^{2}$$
37$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
41$C_1$ $$( 1 - T )^{4}$$
43$C_2^2$ $$1 + T^{4}$$
47$C_2^2$ $$1 + T^{4}$$
53$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
59$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
61$C_2$ $$( 1 + T^{2} )^{2}$$
67$C_2^2$ $$1 + T^{4}$$
71$C_2$ $$( 1 + T^{2} )^{2}$$
73$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
79$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
83$C_2^2$ $$1 + T^{4}$$
89$C_2$ $$( 1 + T^{2} )^{2}$$
97$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}