# Properties

 Degree 6 Conductor $2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s − 3·3-s + 6·4-s − 9·6-s + 10·8-s + 6·9-s + 3·11-s − 18·12-s + 15·16-s − 3·17-s + 18·18-s + 3·19-s + 9·22-s + 9·23-s − 30·24-s − 6·25-s − 10·27-s + 9·29-s − 6·31-s + 21·32-s − 9·33-s − 9·34-s + 36·36-s + 9·37-s + 9·38-s − 12·41-s + 9·43-s + ⋯
 L(s)  = 1 + 2.12·2-s − 1.73·3-s + 3·4-s − 3.67·6-s + 3.53·8-s + 2·9-s + 0.904·11-s − 5.19·12-s + 15/4·16-s − 0.727·17-s + 4.24·18-s + 0.688·19-s + 1.91·22-s + 1.87·23-s − 6.12·24-s − 6/5·25-s − 1.92·27-s + 1.67·29-s − 1.07·31-s + 3.71·32-s − 1.56·33-s − 1.54·34-s + 6·36-s + 1.47·37-s + 1.45·38-s − 1.87·41-s + 1.37·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$6$$ $$N$$ = $$2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{3234} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(6,\ 2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$13.51962232$$ $$L(\frac12)$$ $$\approx$$ $$13.51962232$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7,\;11\}$,$$F_p(T)$$ is a polynomial of degree 6. If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 - T )^{3}$$
3$C_1$ $$( 1 + T )^{3}$$
7 $$1$$
11$C_1$ $$( 1 - T )^{3}$$
good5$S_4\times C_2$ $$1 + 6 T^{2} - 4 T^{3} + 6 p T^{4} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 + 3 T^{2} - 32 T^{3} + 3 p T^{4} + p^{3} T^{6}$$
17$C_2$ $$( 1 + T + p T^{2} )^{3}$$
19$S_4\times C_2$ $$1 - 3 T + 21 T^{2} - 150 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 - 9 T + 72 T^{2} - 393 T^{3} + 72 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 - 9 T + 3 p T^{2} - 430 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 + 6 T - 3 T^{2} - 140 T^{3} - 3 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - 9 T + 99 T^{2} - 502 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 + 12 T + 144 T^{2} + 22 p T^{3} + 144 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 9 T + 117 T^{2} - 610 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 - 15 T + 192 T^{2} - 1391 T^{3} + 192 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 + 123 T^{2} + 32 T^{3} + 123 p T^{4} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 + 9 T + 81 T^{2} + 294 T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 6 T + 54 T^{2} - 506 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 - 6 T + 186 T^{2} - 720 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 - 3 T + 189 T^{2} - 362 T^{3} + 189 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 123 T^{2} + 192 T^{3} + 123 p T^{4} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 - 12 T + 204 T^{2} - 1528 T^{3} + 204 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 6 T + 138 T^{2} - 1184 T^{3} + 138 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 - 36 T + 663 T^{2} - 7736 T^{3} + 663 p T^{4} - 36 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 + 9 T + 210 T^{2} + 1753 T^{3} + 210 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}