# Properties

 Degree 8 Conductor $2^{20} \cdot 7^{4}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 18·5-s + 7·9-s + 64·13-s + 14·17-s + 131·25-s − 128·29-s + 2·37-s − 160·41-s − 126·45-s + 94·49-s − 14·53-s − 158·61-s − 1.15e3·65-s − 286·73-s + 81·81-s − 252·85-s + 194·89-s − 352·97-s + 30·101-s − 226·109-s + 224·113-s + 448·117-s − 233·121-s − 342·125-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 3.59·5-s + 7/9·9-s + 4.92·13-s + 0.823·17-s + 5.23·25-s − 4.41·29-s + 2/37·37-s − 3.90·41-s − 2.79·45-s + 1.91·49-s − 0.264·53-s − 2.59·61-s − 17.7·65-s − 3.91·73-s + 81-s − 2.96·85-s + 2.17·89-s − 3.62·97-s + 0.297·101-s − 2.07·109-s + 1.98·113-s + 3.82·117-s − 1.92·121-s − 2.73·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{20} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{224} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 2^{20} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.0469269$$ $$L(\frac12)$$ $$\approx$$ $$0.0469269$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
7$C_2^2$ $$1 - 94 T^{2} + p^{4} T^{4}$$
good3$C_2^3$ $$1 - 7 T^{2} - 32 T^{4} - 7 p^{4} T^{6} + p^{8} T^{8}$$
5$C_2^2$ $$( 1 + 9 T + 56 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
11$C_2^3$ $$1 + 233 T^{2} + 39648 T^{4} + 233 p^{4} T^{6} + p^{8} T^{8}$$
13$C_2$ $$( 1 - 16 T + p^{2} T^{2} )^{4}$$
17$C_2^2$ $$( 1 - 7 T - 240 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
19$C_2^2$$\times$$C_2^2$ $$( 1 - 46 T^{2} + p^{4} T^{4} )( 1 + 647 T^{2} + p^{4} T^{4} )$$
23$C_2^3$ $$1 + 697 T^{2} + 205968 T^{4} + 697 p^{4} T^{6} + p^{8} T^{8}$$
29$C_2$ $$( 1 + 32 T + p^{2} T^{2} )^{4}$$
31$C_2^3$ $$1 + 1801 T^{2} + 2320080 T^{4} + 1801 p^{4} T^{6} + p^{8} T^{8}$$
37$C_2^2$ $$( 1 - T - 1368 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$C_2$ $$( 1 + 40 T + p^{2} T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 2098 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^3$ $$1 - 2807 T^{2} + 2999568 T^{4} - 2807 p^{4} T^{6} + p^{8} T^{8}$$
53$C_2^2$ $$( 1 + 7 T - 2760 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
59$C_2^3$ $$1 + 4153 T^{2} + 5130048 T^{4} + 4153 p^{4} T^{6} + p^{8} T^{8}$$
61$C_2^2$ $$( 1 + 79 T + 2520 T^{2} + 79 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$C_2^3$ $$1 + 8857 T^{2} + 58295328 T^{4} + 8857 p^{4} T^{6} + p^{8} T^{8}$$
71$C_2^2$ $$( 1 - 7778 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2$ $$( 1 + 46 T + p^{2} T^{2} )^{2}( 1 + 97 T + p^{2} T^{2} )^{2}$$
79$C_2^3$ $$1 + 11257 T^{2} + 87769968 T^{4} + 11257 p^{4} T^{6} + p^{8} T^{8}$$
83$C_2^2$ $$( 1 - 13714 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 97 T + 1488 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
97$C_2$ $$( 1 + 88 T + p^{2} T^{2} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}