# Properties

 Degree 8 Conductor $2^{16} \cdot 3^{12} \cdot 7^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·7-s − 14·25-s + 20·37-s − 20·43-s − 2·49-s − 8·67-s + 52·79-s − 28·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 56·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
 L(s)  = 1 − 1.51·7-s − 2.79·25-s + 3.28·37-s − 3.04·43-s − 2/7·49-s − 0.977·67-s + 5.85·79-s − 2.68·109-s + 3.63·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 4.23·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{16} \cdot 3^{12} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{3024} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$1.610633978$$ $$L(\frac12)$$ $$\approx$$ $$1.610633978$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3 $$1$$
7$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
good5$C_2^2$ $$( 1 + 7 T^{2} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 20 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 20 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + 7 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 16 T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 38 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 8 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 56 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 - 5 T + p T^{2} )^{4}$$
41$C_2^2$ $$( 1 + 7 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + 5 T + p T^{2} )^{4}$$
47$C_2^2$ $$( 1 + 19 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 22 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 43 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 116 T^{2} + p^{2} T^{4} )^{2}$$
67$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
71$C_2^2$ $$( 1 + 58 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2$ $$( 1 - p T^{2} )^{4}$$
79$C_2$ $$( 1 - 13 T + p T^{2} )^{4}$$
83$C_2^2$ $$( 1 + 163 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 70 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 + 100 T^{2} + p^{2} T^{4} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}