# Properties

 Degree 6 Conductor $3^{3} \cdot 7^{6} \cdot 41^{3}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 3

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 3·3-s − 4-s − 4·5-s + 3·6-s − 8-s + 6·9-s − 4·10-s − 4·11-s − 3·12-s − 8·13-s − 12·15-s − 16-s − 2·17-s + 6·18-s − 2·19-s + 4·20-s − 4·22-s − 10·23-s − 3·24-s + 3·25-s − 8·26-s + 10·27-s − 6·29-s − 12·30-s + 2·31-s − 32-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.78·5-s + 1.22·6-s − 0.353·8-s + 2·9-s − 1.26·10-s − 1.20·11-s − 0.866·12-s − 2.21·13-s − 3.09·15-s − 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.458·19-s + 0.894·20-s − 0.852·22-s − 2.08·23-s − 0.612·24-s + 3/5·25-s − 1.56·26-s + 1.92·27-s − 1.11·29-s − 2.19·30-s + 0.359·31-s − 0.176·32-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$6$$ $$N$$ = $$3^{3} \cdot 7^{6} \cdot 41^{3}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : induced by $\chi_{6027} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 3 Selberg data = $(6,\ 3^{3} \cdot 7^{6} \cdot 41^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;7,\;41\}$, $$F_p(T)$$ is a polynomial of degree 6. If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ $$( 1 - T )^{3}$$
7 $$1$$
41$C_1$ $$( 1 + T )^{3}$$
good2$S_4\times C_2$ $$1 - T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
5$S_4\times C_2$ $$1 + 4 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 + 4 T + 34 T^{2} + 84 T^{3} + 34 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 + 8 T + 53 T^{2} + 204 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 + 2 T + 28 T^{2} + 6 T^{3} + 28 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 + 2 T + 51 T^{2} + 68 T^{3} + 51 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 + 10 T + 95 T^{2} + 476 T^{3} + 95 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 + 6 T + 60 T^{2} + 262 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 - 2 T + 2 T^{2} + 132 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - 20 T + 228 T^{2} - 1646 T^{3} + 228 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 10 T + 10 T^{2} + 296 T^{3} + 10 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 4 T + 106 T^{2} + 384 T^{3} + 106 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 - 14 T + 3 p T^{2} - 1452 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 - 8 T + 137 T^{2} - 976 T^{3} + 137 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 8 T + 188 T^{2} - 930 T^{3} + 188 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 - 12 T + 77 T^{2} - 632 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 + 32 T + 550 T^{2} + 5712 T^{3} + 550 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 4 T + 120 T^{2} + 130 T^{3} + 120 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 + 20 T + 305 T^{2} + 3192 T^{3} + 305 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 14 T + 259 T^{2} - 2028 T^{3} + 259 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 + 14 T + 263 T^{2} + 2308 T^{3} + 263 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - 12 T + 305 T^{2} - 2180 T^{3} + 305 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}