# Properties

 Degree 6 Conductor $2^{18} \cdot 3^{3} \cdot 7^{6}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 3

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·3-s + 6·9-s + 3·13-s + 6·17-s − 3·19-s − 6·23-s − 6·25-s − 10·27-s − 12·29-s + 3·31-s − 3·37-s − 9·39-s − 6·41-s − 15·43-s + 12·47-s − 18·51-s − 6·53-s + 9·57-s − 12·59-s + 18·61-s − 9·67-s + 18·69-s + 33·73-s + 18·75-s − 27·79-s + 15·81-s − 18·83-s + ⋯
 L(s)  = 1 − 1.73·3-s + 2·9-s + 0.832·13-s + 1.45·17-s − 0.688·19-s − 1.25·23-s − 6/5·25-s − 1.92·27-s − 2.22·29-s + 0.538·31-s − 0.493·37-s − 1.44·39-s − 0.937·41-s − 2.28·43-s + 1.75·47-s − 2.52·51-s − 0.824·53-s + 1.19·57-s − 1.56·59-s + 2.30·61-s − 1.09·67-s + 2.16·69-s + 3.86·73-s + 2.07·75-s − 3.03·79-s + 5/3·81-s − 1.97·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$6$$ $$N$$ = $$2^{18} \cdot 3^{3} \cdot 7^{6}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : induced by $\chi_{9408} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$3$$ Selberg data = $$(6,\ 2^{18} \cdot 3^{3} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 6. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3$C_1$ $$( 1 + T )^{3}$$
7 $$1$$
good5$S_4\times C_2$ $$1 + 6 T^{2} - 4 T^{3} + 6 p T^{4} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 + 6 T^{2} - 38 T^{3} + 6 p T^{4} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 - 3 T + 3 T^{2} + 34 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 - 6 T + 27 T^{2} - 108 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 + 3 T + 21 T^{2} + 2 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 + 6 T + 45 T^{2} + 180 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 + 12 T + 126 T^{2} + 728 T^{3} + 126 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 - 3 T + 72 T^{2} - 139 T^{3} + 72 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 + 3 T + 27 T^{2} - 146 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 + 6 T + 27 T^{2} - 20 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 + 15 T + 177 T^{2} + 1318 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 - 12 T + 153 T^{2} - 1016 T^{3} + 153 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 + 6 T + 78 T^{2} + 802 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 + 12 T + 198 T^{2} + 1334 T^{3} + 198 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 18 T + 195 T^{2} - 1644 T^{3} + 195 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 + 9 T + 189 T^{2} + 1042 T^{3} + 189 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 + 177 T^{2} - 32 T^{3} + 177 p T^{4} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 - 33 T + 555 T^{2} - 5814 T^{3} + 555 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 + 27 T + 432 T^{2} + 4619 T^{3} + 432 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 + 18 T + 234 T^{2} + 2040 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 + 12 T + 279 T^{2} + 2024 T^{3} + 279 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 + 264 T^{2} + 38 T^{3} + 264 p T^{4} + p^{3} T^{6}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}