# Properties

 Degree 8 Conductor $2^{24} \cdot 3^{12}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·13-s + 10·25-s + 16·37-s − 2·49-s + 16·61-s + 20·73-s + 44·97-s + 16·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 − 2.21·13-s + 2·25-s + 2.63·37-s − 2/7·49-s + 2.04·61-s + 2.34·73-s + 4.46·97-s + 1.53·109-s − 3.45·121-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 − 0.923·169-s + 0.0760·173-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 + 0.0688·211-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{24} \cdot 3^{12}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{1728} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$2.169909677$$ $$L(\frac12)$$ $$\approx$$ $$2.169909677$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
7$C_2^2$ $$( 1 + T^{2} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 19 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
17$C_2^2$ $$( 1 - 14 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2$ $$( 1 - p T^{2} )^{4}$$
23$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 38 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 47 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
41$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 26 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 82 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 101 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 106 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 74 T^{2} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 34 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2$ $$( 1 - 5 T + p T^{2} )^{4}$$
79$C_2$ $$( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2}$$
83$C_2^2$ $$( 1 + 19 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 158 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 - 11 T + p T^{2} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}