# Properties

 Degree 4 Conductor $5^{2} \cdot 31^{2}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4-s + 5-s − 2·9-s + 2·19-s − 20-s − 2·31-s + 2·36-s − 2·41-s − 2·45-s − 49-s − 2·59-s + 64-s + 2·71-s − 2·76-s + 3·81-s + 2·95-s + 2·101-s + 2·109-s + 2·121-s + 2·124-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
 L(s)  = 1 − 4-s + 5-s − 2·9-s + 2·19-s − 20-s − 2·31-s + 2·36-s − 2·41-s − 2·45-s − 49-s − 2·59-s + 64-s + 2·71-s − 2·76-s + 3·81-s + 2·95-s + 2·101-s + 2·109-s + 2·121-s + 2·124-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$24025$$    =    $$5^{2} \cdot 31^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{155} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 24025,\ (\ :0, 0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.3694422763$ $L(\frac12)$ $\approx$ $0.3694422763$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;31\}$, $$F_p$$ is a polynomial of degree 4. If $p \in \{5,\;31\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ $$1 - T + T^{2}$$
31$C_1$ $$( 1 + T )^{2}$$
good2$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
3$C_2$ $$( 1 + T^{2} )^{2}$$
7$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
11$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
13$C_2$ $$( 1 + T^{2} )^{2}$$
17$C_2$ $$( 1 + T^{2} )^{2}$$
19$C_2$ $$( 1 - T + T^{2} )^{2}$$
23$C_2$ $$( 1 + T^{2} )^{2}$$
29$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
37$C_2$ $$( 1 + T^{2} )^{2}$$
41$C_2$ $$( 1 + T + T^{2} )^{2}$$
43$C_2$ $$( 1 + T^{2} )^{2}$$
47$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
53$C_2$ $$( 1 + T^{2} )^{2}$$
59$C_2$ $$( 1 + T + T^{2} )^{2}$$
61$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
71$C_2$ $$( 1 - T + T^{2} )^{2}$$
73$C_2$ $$( 1 + T^{2} )^{2}$$
79$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
83$C_2$ $$( 1 + T^{2} )^{2}$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
97$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}