# Properties

 Degree 4 Conductor $7 \cdot 11^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s − 4-s − 5-s − 6-s − 3·7-s + 8-s − 3·9-s + 10-s + 2·11-s − 12-s + 8·13-s + 3·14-s − 15-s − 16-s + 2·17-s + 3·18-s + 20-s − 3·21-s − 2·22-s − 5·23-s + 24-s − 7·25-s − 8·26-s − 4·27-s + 3·28-s − 6·29-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 2.21·13-s + 0.801·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.223·20-s − 0.654·21-s − 0.426·22-s − 1.04·23-s + 0.204·24-s − 7/5·25-s − 1.56·26-s − 0.769·27-s + 0.566·28-s − 1.11·29-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$847$$    =    $$7 \cdot 11^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{847} (1, \cdot )$ Sato-Tate : $G_{3,3}$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 847,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.3365454249$ $L(\frac12)$ $\approx$ $0.3365454249$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{7,\;11\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 2 T + p T^{2} )$$
11$C_1$ $$( 1 - T )^{2}$$
good2$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + p T + p T^{2} )$$
3$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} )$$
5$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
17$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
43$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
53$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
59$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
89$C_2$$\times$$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}