# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·3-s − 3·4-s + 14·7-s + 9·9-s + 12·12-s − 8·13-s + 16·16-s − 32·19-s − 56·21-s − 45·25-s − 44·27-s − 42·28-s + 6·31-s − 27·36-s − 24·37-s + 32·39-s + 176·43-s − 64·48-s + 49·49-s + 24·52-s + 128·57-s + 52·61-s + 126·63-s − 117·64-s − 104·67-s − 36·73-s + 180·75-s + ⋯
 L(s)  = 1 − 4/3·3-s − 3/4·4-s + 2·7-s + 9-s + 12-s − 0.615·13-s + 16-s − 1.68·19-s − 8/3·21-s − 9/5·25-s − 1.62·27-s − 3/2·28-s + 6/31·31-s − 3/4·36-s − 0.648·37-s + 0.820·39-s + 4.09·43-s − 4/3·48-s + 49-s + 6/13·52-s + 2.24·57-s + 0.852·61-s + 2·63-s − 1.82·64-s − 1.55·67-s − 0.493·73-s + 12/5·75-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$194481$$    =    $$3^{4} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{21} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 194481,\ (\ :1, 1, 1, 1),\ 1)$ $L(\frac{3}{2})$ $\approx$ $0.422485$ $L(\frac12)$ $\approx$ $0.422485$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;7\}$, $$F_p$$ is a polynomial of degree 8. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3$C_2^2$ $$1 + 4 T + 7 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4}$$
7$C_2$ $$( 1 - p T + p^{2} T^{2} )^{2}$$
good2$C_2^3$ $$1 + 3 T^{2} - 7 T^{4} + 3 p^{4} T^{6} + p^{8} T^{8}$$
5$C_2^3$ $$1 + 9 p T^{2} + 56 p^{2} T^{4} + 9 p^{5} T^{6} + p^{8} T^{8}$$
11$C_2^3$ $$1 + 117 T^{2} - 952 T^{4} + 117 p^{4} T^{6} + p^{8} T^{8}$$
13$C_2$ $$( 1 + 2 T + p^{2} T^{2} )^{4}$$
17$C_2^3$ $$1 - 142 T^{2} - 63357 T^{4} - 142 p^{4} T^{6} + p^{8} T^{8}$$
19$C_2^2$ $$( 1 + 16 T - 105 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$C_2^2$$\times$$C_2^2$ $$( 1 - 44 T + 1407 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )( 1 + 44 T + 1407 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )$$
29$C_2^2$ $$( 1 - 1437 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 3 T - 952 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 12 T - 1225 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 2382 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2$ $$( 1 - 44 T + p^{2} T^{2} )^{4}$$
47$C_2^3$ $$1 + 4238 T^{2} + 13080963 T^{4} + 4238 p^{4} T^{6} + p^{8} T^{8}$$
53$C_2^3$ $$1 + 5213 T^{2} + 19284888 T^{4} + 5213 p^{4} T^{6} + p^{8} T^{8}$$
59$C_2^3$ $$1 + 6557 T^{2} + 30876888 T^{4} + 6557 p^{4} T^{6} + p^{8} T^{8}$$
61$C_2^2$ $$( 1 - 26 T - 3045 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 52 T - 1785 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 1262 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 18 T - 5005 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$C_1$$\times$$C_2$ $$( 1 - p T )^{4}( 1 + p T + p^{2} T^{2} )^{2}$$
83$C_2^2$ $$( 1 + 6067 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^3$ $$1 + 13422 T^{2} + 117407843 T^{4} + 13422 p^{4} T^{6} + p^{8} T^{8}$$
97$C_2$ $$( 1 + 93 T + p^{2} T^{2} )^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}