# Properties

 Degree 6 Conductor $71^{3}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 5-s + 6-s + 10-s + 15-s − 19-s − 29-s − 30-s − 37-s + 38-s − 43-s + 3·49-s + 57-s + 58-s + 3·71-s − 73-s + 74-s − 79-s − 83-s + 86-s + 87-s − 89-s + 95-s − 3·98-s − 101-s − 103-s + ⋯
 L(s)  = 1 − 2-s − 3-s − 5-s + 6-s + 10-s + 15-s − 19-s − 29-s − 30-s − 37-s + 38-s − 43-s + 3·49-s + 57-s + 58-s + 3·71-s − 73-s + 74-s − 79-s − 83-s + 86-s + 87-s − 89-s + 95-s − 3·98-s − 101-s − 103-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 357911 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 357911 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$6$$ $$N$$ = $$357911$$    =    $$71^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{71} (70, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(6,\ 357911,\ (\ :0, 0, 0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.05498223480$ $L(\frac12)$ $\approx$ $0.05498223480$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 71$, $$F_p(T)$$ is a polynomial of degree 6. If $p = 71$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad71$C_1$ $$( 1 - T )^{3}$$
good2$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
3$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
5$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
7$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
11$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
13$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
17$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
19$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
23$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
29$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
31$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
37$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
41$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
43$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
47$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
59$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
61$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
73$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
79$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
83$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
89$C_6$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
97$C_1$$\times$$C_1$ $$( 1 - T )^{3}( 1 + T )^{3}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}