# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 4·3-s + 2·4-s − 2·5-s + 8·6-s − 4·7-s + 7·9-s + 4·10-s − 8·12-s + 4·13-s + 8·14-s + 8·15-s − 4·16-s − 8·17-s − 14·18-s + 4·19-s − 4·20-s + 16·21-s − 3·25-s − 8·26-s − 4·27-s − 8·28-s − 8·29-s − 16·30-s + 8·32-s + 16·34-s + 8·35-s + ⋯
 L(s)  = 1 − 1.41·2-s − 2.30·3-s + 4-s − 0.894·5-s + 3.26·6-s − 1.51·7-s + 7/3·9-s + 1.26·10-s − 2.30·12-s + 1.10·13-s + 2.13·14-s + 2.06·15-s − 16-s − 1.94·17-s − 3.29·18-s + 0.917·19-s − 0.894·20-s + 3.49·21-s − 3/5·25-s − 1.56·26-s − 0.769·27-s − 1.51·28-s − 1.48·29-s − 2.92·30-s + 1.41·32-s + 2.74·34-s + 1.35·35-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$968$$    =    $$2^{3} \cdot 11^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{968} (1, \cdot )$ Sato-Tate : $G_{3,3}$ primitive : no self-dual : yes analytic rank = 1 Selberg data = $(4,\ 968,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;11\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 + p T + p T^{2}$$
11$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good3$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + p T + p T^{2} )$$
5$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} )$$
7$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
13$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
17$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
23$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 8 T + p T^{2} )$$
31$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
43$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
71$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
89$C_2$ $$( 1 - 15 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}