# Properties

 Degree 8 Conductor $2^{24} \cdot 3^{8} \cdot 7^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 20·25-s − 24·43-s − 2·49-s + 48·67-s − 8·73-s − 40·97-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
 L(s)  = 1 − 4·25-s − 3.65·43-s − 2/7·49-s + 5.86·67-s − 0.936·73-s − 4.06·97-s + 8/11·121-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 − 1.53·169-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 + 0.0669·223-s + 0.0663·227-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \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^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{24} \cdot 3^{8} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{4032} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 2^{24} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$ $L(1)$ $\approx$ $0.5695052350$ $L(\frac12)$ $\approx$ $0.5695052350$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $$F_p$$ is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2 $$1$$
3 $$1$$
7$C_2$ $$( 1 + T^{2} )^{2}$$
good5$C_2$ $$( 1 + p T^{2} )^{4}$$
11$C_2^2$ $$( 1 - 4 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - p T^{2} )^{4}$$
19$C_2$ $$( 1 + p T^{2} )^{4}$$
23$C_2^2$ $$( 1 + 28 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 40 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 46 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 38 T^{2} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
47$C_2^2$ $$( 1 + 22 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 56 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 46 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 86 T^{2} + p^{2} T^{4} )^{2}$$
67$C_2$ $$( 1 - 12 T + p T^{2} )^{4}$$
71$C_2^2$ $$( 1 + 124 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 58 T^{2} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 122 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 106 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 + 10 T + p T^{2} )^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}