# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s − 4·5-s + 4·7-s + 6·9-s + 4·11-s − 12·13-s − 16·15-s + 4·19-s + 16·21-s + 8·25-s − 4·27-s − 12·29-s + 16·33-s − 16·35-s − 4·37-s − 48·39-s − 24·41-s + 4·43-s − 24·45-s + 10·49-s − 12·53-s − 16·55-s + 16·57-s + 12·59-s + 4·61-s + 24·63-s + 48·65-s + ⋯
 L(s)  = 1 + 2.30·3-s − 1.78·5-s + 1.51·7-s + 2·9-s + 1.20·11-s − 3.32·13-s − 4.13·15-s + 0.917·19-s + 3.49·21-s + 8/5·25-s − 0.769·27-s − 2.22·29-s + 2.78·33-s − 2.70·35-s − 0.657·37-s − 7.68·39-s − 3.74·41-s + 0.609·43-s − 3.57·45-s + 10/7·49-s − 1.64·53-s − 2.15·55-s + 2.11·57-s + 1.56·59-s + 0.512·61-s + 3.02·63-s + 5.95·65-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{24} \cdot 3^{4} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{1344} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 2^{24} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$ $L(1)$ $\approx$ $2.248486461$ $L(\frac12)$ $\approx$ $2.248486461$ $L(\frac{3}{2})$ 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$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
7$C_1$ $$( 1 - T )^{4}$$
good5$D_4\times C_2$ $$1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 - 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
13$D_4\times C_2$ $$1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 324 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
19$D_4\times C_2$ $$1 - 4 T + 8 T^{2} - 68 T^{3} + 574 T^{4} - 68 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 - 4 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2}$$
31$C_4\times C_2$ $$1 + 28 T^{2} + 966 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 92 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
41$C_4$ $$( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 4 T + 8 T^{2} - 116 T^{3} + 1486 T^{4} - 116 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2^2$ $$( 1 + 62 T^{2} + p^{2} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 + 12 T + 72 T^{2} + 660 T^{3} + 6046 T^{4} + 660 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2$$\times$$C_2^2$ $$( 1 - 6 T + p T^{2} )^{2}( 1 - 82 T^{2} + p^{2} T^{4} )$$
61$D_4\times C_2$ $$1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 236 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^2$ $$( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 106 T^{2} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 148 T^{2} + 14086 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2^2$ $$( 1 - 154 T^{2} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 4 T + 8 T^{2} + 444 T^{3} - 12994 T^{4} + 444 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}