# Properties

 Degree 4 Conductor $2^{7} \cdot 5^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s + 4·7-s − 8-s − 5·9-s − 4·14-s + 16-s − 6·17-s + 5·18-s + 12·23-s + 4·28-s + 4·31-s − 32-s + 6·34-s − 5·36-s − 6·41-s − 12·46-s + 24·47-s − 2·49-s − 4·56-s − 4·62-s − 20·63-s + 64-s − 6·68-s + 24·71-s + 5·72-s + 22·73-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 5/3·9-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 1.17·18-s + 2.50·23-s + 0.755·28-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 5/6·36-s − 0.937·41-s − 1.76·46-s + 3.50·47-s − 2/7·49-s − 0.534·56-s − 0.508·62-s − 2.51·63-s + 1/8·64-s − 0.727·68-s + 2.84·71-s + 0.589·72-s + 2.57·73-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$80000$$    =    $$2^{7} \cdot 5^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{80000} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 80000,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $1.078912628$ $L(\frac12)$ $\approx$ $1.078912628$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ $$1 + T$$
5 $$1$$
good3$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
7$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
23$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
41$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
71$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 11 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
89$C_2$ $$( 1 - 15 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}