# Properties

 Degree 8 Conductor $2^{28}$ Sign $1$ Motivic weight 4 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 132·9-s − 648·17-s + 2.37e3·25-s + 7.56e3·41-s − 2.68e3·49-s + 1.10e4·73-s − 54·81-s + 9.72e3·89-s + 2.98e4·97-s − 2.80e3·113-s − 3.53e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8.55e4·153-s + 157-s + 163-s + 167-s + 2.09e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 1.62·9-s − 2.24·17-s + 3.79·25-s + 4.49·41-s − 1.11·49-s + 2.06·73-s − 0.00823·81-s + 1.22·89-s + 3.16·97-s − 0.219·113-s − 2.41·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 3.65·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 0.732·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{28}$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : induced by $\chi_{128} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 2^{28} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )$$ $$L(\frac{5}{2})$$ $$\approx$$ $$2.39195$$ $$L(\frac12)$$ $$\approx$$ $$2.39195$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 8. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
good3$C_2^2$ $$( 1 + 22 p T^{2} + p^{8} T^{4} )^{2}$$
5$C_2^2$ $$( 1 - 1186 T^{2} + p^{8} T^{4} )^{2}$$
7$C_2^2$ $$( 1 + 1342 T^{2} + p^{8} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 146 p^{2} T^{2} + p^{8} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 10466 T^{2} + p^{8} T^{4} )^{2}$$
17$C_2$ $$( 1 + 162 T + p^{4} T^{2} )^{4}$$
19$C_2^2$ $$( 1 + 66242 T^{2} + p^{8} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 62018 T^{2} + p^{8} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 285854 T^{2} + p^{8} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 1453826 T^{2} + p^{8} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 1462178 T^{2} + p^{8} T^{4} )^{2}$$
41$C_2$ $$( 1 - 1890 T + p^{4} T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 1630462 T^{2} + p^{8} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 7768706 T^{2} + p^{8} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 11876386 T^{2} + p^{8} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 19112066 T^{2} + p^{8} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 22046306 T^{2} + p^{8} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 37495106 T^{2} + p^{8} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 9393982 T^{2} + p^{8} T^{4} )^{2}$$
73$C_2$ $$( 1 - 2750 T + p^{4} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 13977986 T^{2} + p^{8} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 7728578 T^{2} + p^{8} T^{4} )^{2}$$
89$C_2$ $$( 1 - 2430 T + p^{4} T^{2} )^{4}$$
97$C_2$ $$( 1 - 7454 T + p^{4} T^{2} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}