# Properties

 Degree 6 Conductor $2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 3

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s + 6·5-s − 3·7-s − 9-s − 3·11-s + 3·13-s + 12·15-s − 13·17-s + 19-s − 6·21-s − 13·23-s + 13·25-s − 7·27-s + 8·29-s − 17·31-s − 6·33-s − 18·35-s + 7·37-s + 6·39-s − 10·41-s + 9·43-s − 6·45-s − 2·47-s + 6·49-s − 26·51-s − 11·53-s − 18·55-s + ⋯
 L(s)  = 1 + 1.15·3-s + 2.68·5-s − 1.13·7-s − 1/3·9-s − 0.904·11-s + 0.832·13-s + 3.09·15-s − 3.15·17-s + 0.229·19-s − 1.30·21-s − 2.71·23-s + 13/5·25-s − 1.34·27-s + 1.48·29-s − 3.05·31-s − 1.04·33-s − 3.04·35-s + 1.15·37-s + 0.960·39-s − 1.56·41-s + 1.37·43-s − 0.894·45-s − 0.291·47-s + 6/7·49-s − 3.64·51-s − 1.51·53-s − 2.42·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$6$$ $$N$$ = $$2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : induced by $\chi_{8008} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 3 Selberg data = $(6,\ 2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;7,\;11,\;13\}$, $$F_p$$ is a polynomial of degree 6. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad2 $$1$$
7$C_1$ $$( 1 + T )^{3}$$
11$C_1$ $$( 1 + T )^{3}$$
13$C_1$ $$( 1 - T )^{3}$$
good3$S_4\times C_2$ $$1 - 2 T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
5$S_4\times C_2$ $$1 - 6 T + 23 T^{2} - 59 T^{3} + 23 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 + 13 T + 88 T^{2} + 414 T^{3} + 88 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 - T + 42 T^{2} - 10 T^{3} + 42 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 + 13 T + 75 T^{2} + 330 T^{3} + 75 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 - 8 T + 59 T^{2} - 256 T^{3} + 59 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 + 17 T + 183 T^{2} + 1202 T^{3} + 183 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - 7 T + 121 T^{2} - 514 T^{3} + 121 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 + 10 T - T^{2} - 364 T^{3} - p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 9 T + 120 T^{2} - 720 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 2 T + 81 T^{2} - 36 T^{3} + 81 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 + 11 T + 98 T^{2} + 712 T^{3} + 98 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 - 9 T + 147 T^{2} - 866 T^{3} + 147 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 9 T + 206 T^{2} - 1114 T^{3} + 206 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 + 10 T + 219 T^{2} + 1303 T^{3} + 219 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 + 22 T + 5 p T^{2} + 3351 T^{3} + 5 p^{2} T^{4} + 22 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 18 T + 263 T^{2} + 2524 T^{3} + 263 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 + 3 T + 2 p T^{2} + 286 T^{3} + 2 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 + T + 168 T^{2} - 108 T^{3} + 168 p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 - 16 T + 295 T^{2} - 2607 T^{3} + 295 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - T + 35 T^{2} - 630 T^{3} + 35 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}