# Properties

 Degree 4 Conductor $7^{2} \cdot 11^{3}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·7-s − 5·9-s + 11-s + 8·13-s − 4·16-s − 4·17-s − 2·23-s − 9·25-s + 6·37-s − 16·41-s − 3·49-s − 12·53-s + 24·61-s + 10·63-s − 14·67-s − 6·71-s + 8·73-s − 2·77-s + 16·81-s − 12·83-s − 16·91-s − 5·99-s + 4·101-s + 8·112-s + 18·113-s − 40·117-s + 8·119-s + ⋯
 L(s)  = 1 − 0.755·7-s − 5/3·9-s + 0.301·11-s + 2.21·13-s − 16-s − 0.970·17-s − 0.417·23-s − 9/5·25-s + 0.986·37-s − 2.49·41-s − 3/7·49-s − 1.64·53-s + 3.07·61-s + 1.25·63-s − 1.71·67-s − 0.712·71-s + 0.936·73-s − 0.227·77-s + 16/9·81-s − 1.31·83-s − 1.67·91-s − 0.502·99-s + 0.398·101-s + 0.755·112-s + 1.69·113-s − 3.69·117-s + 0.733·119-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$65219$$    =    $$7^{2} \cdot 11^{3}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{65219} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 1 Selberg data = $(4,\ 65219,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ 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$C_2$ $$1 + 2 T + p T^{2}$$
11$C_1$ $$1 - T$$
good2$C_2$ $$( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )$$
3$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
5$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
13$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$C_2$ $$( 1 + T + p T^{2} )^{2}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
37$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
61$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} )$$
97$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}