# Properties

 Degree 4 Conductor $47^{2}$ 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 + 6-s − 7-s + 14-s − 17-s + 21-s + 2·25-s + 32-s + 34-s − 37-s − 42-s + 2·47-s − 2·50-s + 51-s − 53-s − 59-s − 61-s − 64-s − 71-s + 74-s − 2·75-s − 79-s + 4·83-s − 89-s − 2·94-s − 96-s + ⋯
 L(s)  = 1 − 2-s − 3-s + 6-s − 7-s + 14-s − 17-s + 21-s + 2·25-s + 32-s + 34-s − 37-s − 42-s + 2·47-s − 2·50-s + 51-s − 53-s − 59-s − 61-s − 64-s − 71-s + 74-s − 2·75-s − 79-s + 4·83-s − 89-s − 2·94-s − 96-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$2209$$    =    $$47^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{47} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 2209,\ (\ :0, 0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.08912006073$ $L(\frac12)$ $\approx$ $0.08912006073$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 47$, $$F_p$$ is a polynomial of degree 4. If $p = 47$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad47$C_1$ $$( 1 - T )^{2}$$
good2$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
3$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
5$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
7$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
11$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
13$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
17$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
23$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
29$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
31$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
37$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
41$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
53$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
59$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
61$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
71$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
73$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
79$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
83$C_1$ $$( 1 - T )^{4}$$
89$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
97$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}