# Properties

 Degree $6$ Conductor $2662500456$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s + 6·4-s + 2·5-s + 3·7-s + 10·8-s + 6·10-s + 3·11-s + 9·14-s + 15·16-s + 2·17-s + 10·19-s + 12·20-s + 9·22-s + 2·23-s − 25-s + 18·28-s − 6·29-s + 21·32-s + 6·34-s + 6·35-s + 10·37-s + 30·38-s + 20·40-s − 2·41-s + 14·43-s + 18·44-s + 6·46-s + ⋯
 L(s)  = 1 + 2.12·2-s + 3·4-s + 0.894·5-s + 1.13·7-s + 3.53·8-s + 1.89·10-s + 0.904·11-s + 2.40·14-s + 15/4·16-s + 0.485·17-s + 2.29·19-s + 2.68·20-s + 1.91·22-s + 0.417·23-s − 1/5·25-s + 3.40·28-s − 1.11·29-s + 3.71·32-s + 1.02·34-s + 1.01·35-s + 1.64·37-s + 4.86·38-s + 3.16·40-s − 0.312·41-s + 2.13·43-s + 2.71·44-s + 0.884·46-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$6$$ Conductor: $$2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 11^{3}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1386} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$26.27586764$$ $$L(\frac12)$$ $$\approx$$ $$26.27586764$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 - T )^{3}$$
3 $$1$$
7$C_1$ $$( 1 - T )^{3}$$
11$C_1$ $$( 1 - T )^{3}$$
good5$S_4\times C_2$ $$1 - 2 T + p T^{2} - 8 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 - 5 T^{2} - 16 T^{3} - 5 p T^{4} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 - 2 T + 41 T^{2} - 56 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 - 10 T + 51 T^{2} - 192 T^{3} + 51 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 - 2 T + 13 T^{2} - 44 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 + 6 T + 55 T^{2} + 252 T^{3} + 55 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 - 9 T^{2} - 128 T^{3} - 9 p T^{4} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - 10 T + 99 T^{2} - 588 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 + 2 T + 113 T^{2} + 152 T^{3} + 113 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 14 T + 73 T^{2} - 228 T^{3} + 73 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 10 T + 103 T^{2} + 544 T^{3} + 103 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 - 14 T + 179 T^{2} - 1412 T^{3} + 179 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 + 8 T + 41 T^{2} - 208 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 8 T + 43 T^{2} + 192 T^{3} + 43 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 - 4 T + 149 T^{2} - 472 T^{3} + 149 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 + 2 T + 53 T^{2} - 580 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 14 T + 273 T^{2} + 2088 T^{3} + 273 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 - 8 T + 201 T^{2} - 1120 T^{3} + 201 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 + 20 T + 371 T^{2} + 3536 T^{3} + 371 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 + 16 T + 295 T^{2} + 2704 T^{3} + 295 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - 18 T + 223 T^{2} - 2524 T^{3} + 223 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$