# Properties

 Degree 4 Conductor $2^{6} \cdot 89$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·4-s + 7-s + 4·9-s + 4·16-s + 6·17-s − 9·23-s − 7·25-s − 2·28-s − 2·31-s − 8·36-s − 3·41-s − 3·47-s − 11·49-s + 4·63-s − 8·64-s − 12·68-s + 15·71-s + 10·73-s − 2·79-s + 7·81-s − 10·89-s + 18·92-s + 16·97-s + 14·100-s − 17·103-s + 4·112-s − 9·113-s + ⋯
 L(s)  = 1 − 4-s + 0.377·7-s + 4/3·9-s + 16-s + 1.45·17-s − 1.87·23-s − 7/5·25-s − 0.377·28-s − 0.359·31-s − 4/3·36-s − 0.468·41-s − 0.437·47-s − 1.57·49-s + 0.503·63-s − 64-s − 1.45·68-s + 1.78·71-s + 1.17·73-s − 0.225·79-s + 7/9·81-s − 1.05·89-s + 1.87·92-s + 1.62·97-s + 7/5·100-s − 1.67·103-s + 0.377·112-s − 0.846·113-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 5696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 5696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$5696$$    =    $$2^{6} \cdot 89$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{5696} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 5696,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.7901956560$ $L(\frac12)$ $\approx$ $0.7901956560$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;89\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;89\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 + p T^{2}$$
89$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 9 T + p T^{2} )$$
good3$V_4$ $$1 - 4 T^{2} + p^{2} T^{4}$$
5$V_4$ $$1 + 7 T^{2} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} )$$
11$V_4$ $$1 - 5 T^{2} + p^{2} T^{4}$$
13$V_4$ $$1 - 8 T^{2} + p^{2} T^{4}$$
17$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
19$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
29$V_4$ $$1 + 40 T^{2} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
37$V_4$ $$1 + 16 T^{2} + p^{2} T^{4}$$
41$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 3 T + p T^{2} )$$
53$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
59$V_4$ $$1 - 8 T^{2} + p^{2} T^{4}$$
61$V_4$ $$1 - 50 T^{2} + p^{2} T^{4}$$
67$V_4$ $$1 + 46 T^{2} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
79$C_2$ $$( 1 + T + p T^{2} )^{2}$$
83$V_4$ $$1 - 50 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
\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}