# Properties

 Degree 4 Conductor $2^{4} \cdot 5^{2} \cdot 7^{2}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s − 3-s − 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 14-s + 15-s − 16-s + 18-s − 21-s − 23-s + 24-s − 2·27-s − 2·29-s + 30-s − 35-s + 40-s − 2·41-s − 42-s + 2·43-s − 45-s − 46-s + 2·47-s + 48-s + ⋯
 L(s)  = 1 + 2-s − 3-s − 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 14-s + 15-s − 16-s + 18-s − 21-s − 23-s + 24-s − 2·27-s − 2·29-s + 30-s − 35-s + 40-s − 2·41-s − 42-s + 2·43-s − 45-s − 46-s + 2·47-s + 48-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$19600$$    =    $$2^{4} \cdot 5^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{140} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 19600,\ (\ :0, 0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.4070629205$ $L(\frac12)$ $\approx$ $0.4070629205$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;7\}$, $$F_p$$ is a polynomial of degree 4. If $p \in \{2,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 - T + T^{2}$$
5$C_2$ $$1 + T + T^{2}$$
7$C_2$ $$1 - T + T^{2}$$
good3$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
11$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
13$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
17$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
19$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
23$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
29$C_2$ $$( 1 + T + T^{2} )^{2}$$
31$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
37$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
41$C_2$ $$( 1 + T + T^{2} )^{2}$$
43$C_2$ $$( 1 - T + T^{2} )^{2}$$
47$C_2$ $$( 1 - T + T^{2} )^{2}$$
53$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
59$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
61$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
67$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
73$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
79$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
83$C_2$ $$( 1 - T + T^{2} )^{2}$$
89$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
97$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}