# Properties

 Degree 6 Conductor $5^{6} \cdot 47^{3}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5·2-s + 5·3-s + 3·4-s + 25·6-s + 45·7-s − 27·8-s − 49·9-s + 2·11-s + 15·12-s + 80·13-s + 225·14-s − 37·16-s + 39·17-s − 245·18-s − 24·19-s + 225·21-s + 10·22-s − 120·23-s − 135·24-s + 400·26-s − 353·27-s + 135·28-s − 184·29-s − 4·31-s + 53·32-s + 10·33-s + 195·34-s + ⋯
 L(s)  = 1 + 1.76·2-s + 0.962·3-s + 3/8·4-s + 1.70·6-s + 2.42·7-s − 1.19·8-s − 1.81·9-s + 0.0548·11-s + 0.360·12-s + 1.70·13-s + 4.29·14-s − 0.578·16-s + 0.556·17-s − 3.20·18-s − 0.289·19-s + 2.33·21-s + 0.0969·22-s − 1.08·23-s − 1.14·24-s + 3.01·26-s − 2.51·27-s + 0.911·28-s − 1.17·29-s − 0.0231·31-s + 0.292·32-s + 0.0527·33-s + 0.983·34-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \,\Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$6$$ $$N$$ = $$5^{6} \cdot 47^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{1175} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(6,\ 5^{6} \cdot 47^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )$ $L(2)$ $\approx$ $18.97284347$ $L(\frac12)$ $\approx$ $18.97284347$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;47\}$, $$F_p$$ is a polynomial of degree 6. If $p \in \{5,\;47\}$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad5 $$1$$
47$C_1$ $$( 1 + p T )^{3}$$
good2$S_4\times C_2$ $$1 - 5 T + 11 p T^{2} - 17 p^{2} T^{3} + 11 p^{4} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6}$$
3$S_4\times C_2$ $$1 - 5 T + 74 T^{2} - 262 T^{3} + 74 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6}$$
7$S_4\times C_2$ $$1 - 45 T + 216 p T^{2} - 32022 T^{3} + 216 p^{4} T^{4} - 45 p^{6} T^{5} + p^{9} T^{6}$$
11$S_4\times C_2$ $$1 - 2 T + 2713 T^{2} + 13108 T^{3} + 2713 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6}$$
13$S_4\times C_2$ $$1 - 80 T + 7775 T^{2} - 355808 T^{3} + 7775 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6}$$
17$S_4\times C_2$ $$1 - 39 T + 10764 T^{2} - 269068 T^{3} + 10764 p^{3} T^{4} - 39 p^{6} T^{5} + p^{9} T^{6}$$
19$S_4\times C_2$ $$1 + 24 T + 19881 T^{2} + 312456 T^{3} + 19881 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6}$$
23$S_4\times C_2$ $$1 + 120 T + 28569 T^{2} + 1922592 T^{3} + 28569 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6}$$
29$S_4\times C_2$ $$1 + 184 T + 39951 T^{2} + 3957616 T^{3} + 39951 p^{3} T^{4} + 184 p^{6} T^{5} + p^{9} T^{6}$$
31$S_4\times C_2$ $$1 + 4 T + 6029 T^{2} + 283720 p T^{3} + 6029 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6}$$
37$S_4\times C_2$ $$1 - 589 T + 7220 p T^{2} - 67144276 T^{3} + 7220 p^{4} T^{4} - 589 p^{6} T^{5} + p^{9} T^{6}$$
41$S_4\times C_2$ $$1 + 92 T + 4027 p T^{2} + 7995456 T^{3} + 4027 p^{4} T^{4} + 92 p^{6} T^{5} + p^{9} T^{6}$$
43$S_4\times C_2$ $$1 - 250 T + 171365 T^{2} - 42748948 T^{3} + 171365 p^{3} T^{4} - 250 p^{6} T^{5} + p^{9} T^{6}$$
53$S_4\times C_2$ $$1 + 459 T + 280872 T^{2} + 63986692 T^{3} + 280872 p^{3} T^{4} + 459 p^{6} T^{5} + p^{9} T^{6}$$
59$S_4\times C_2$ $$1 - 579 T + 228972 T^{2} - 94125566 T^{3} + 228972 p^{3} T^{4} - 579 p^{6} T^{5} + p^{9} T^{6}$$
61$S_4\times C_2$ $$1 - 267 T + 543684 T^{2} - 130418348 T^{3} + 543684 p^{3} T^{4} - 267 p^{6} T^{5} + p^{9} T^{6}$$
67$S_4\times C_2$ $$1 - 540 T + 119601 T^{2} + 142838224 T^{3} + 119601 p^{3} T^{4} - 540 p^{6} T^{5} + p^{9} T^{6}$$
71$S_4\times C_2$ $$1 - 749 T + 628750 T^{2} - 200987390 T^{3} + 628750 p^{3} T^{4} - 749 p^{6} T^{5} + p^{9} T^{6}$$
73$S_4\times C_2$ $$1 - 1924 T + 2114435 T^{2} - 1560841600 T^{3} + 2114435 p^{3} T^{4} - 1924 p^{6} T^{5} + p^{9} T^{6}$$
79$S_4\times C_2$ $$1 - 805 T + 1187566 T^{2} - 671146982 T^{3} + 1187566 p^{3} T^{4} - 805 p^{6} T^{5} + p^{9} T^{6}$$
83$S_4\times C_2$ $$1 + 712 T + 1422097 T^{2} + 602850736 T^{3} + 1422097 p^{3} T^{4} + 712 p^{6} T^{5} + p^{9} T^{6}$$
89$S_4\times C_2$ $$1 - 835 T + 2039856 T^{2} - 1065240352 T^{3} + 2039856 p^{3} T^{4} - 835 p^{6} T^{5} + p^{9} T^{6}$$
97$S_4\times C_2$ $$1 - 2243 T + 3813620 T^{2} - 4231696160 T^{3} + 3813620 p^{3} T^{4} - 2243 p^{6} T^{5} + p^{9} T^{6}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}