# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·9-s + 4·17-s − 6·25-s + 20·41-s − 14·49-s − 12·73-s + 27·81-s + 20·89-s + 36·97-s − 28·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 2·9-s + 0.970·17-s − 6/5·25-s + 3.12·41-s − 2·49-s − 1.40·73-s + 3·81-s + 2.11·89-s + 3.65·97-s − 2.63·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2048 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2048 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$2048$$    =    $$2^{11}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{2048} (1, \cdot )$ Sato-Tate : $N(\mathrm{U}(1))$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 2048,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.6076863142$ $L(\frac12)$ $\approx$ $0.6076863142$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 2$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 $$1$$
good3$C_2$ $$( 1 + p T^{2} )^{2}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
41$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 + p T^{2} )^{2}$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$ $$( 1 + p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 18 T + p T^{2} )^{2}$$
\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}