# Properties

 Degree $6$ Conductor $217298593728$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·11-s + 2·13-s − 6·17-s − 4·19-s + 4·23-s − 11·25-s + 4·31-s + 10·37-s + 14·41-s − 20·43-s − 16·47-s − 5·49-s + 6·53-s + 24·59-s − 2·61-s + 10·67-s + 12·71-s − 10·73-s − 2·79-s + 4·83-s + 12·89-s + 18·97-s − 20·101-s + 6·103-s + 8·107-s − 6·109-s + 20·113-s + ⋯
 L(s)  = 1 + 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s − 2.19·25-s + 0.718·31-s + 1.64·37-s + 2.18·41-s − 3.04·43-s − 2.33·47-s − 5/7·49-s + 0.824·53-s + 3.12·59-s − 0.256·61-s + 1.22·67-s + 1.42·71-s − 1.17·73-s − 0.225·79-s + 0.439·83-s + 1.27·89-s + 1.82·97-s − 1.99·101-s + 0.591·103-s + 0.773·107-s − 0.574·109-s + 1.88·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{6} \cdot 3^{6} \cdot 167^{3}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{6012} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{6} \cdot 3^{6} \cdot 167^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.644573899$$ $$L(\frac12)$$ $$\approx$$ $$1.644573899$$ $$L(\frac{3}{2})$$ 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 $$1$$
3 $$1$$
167$C_1$ $$( 1 + T )^{3}$$
good5$S_4\times C_2$ $$1 + 11 T^{2} + 2 T^{3} + 11 p T^{4} + p^{3} T^{6}$$
7$S_4\times C_2$ $$1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 - 4 T + 3 p T^{2} - 84 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
13$D_{6}$ $$1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 + 6 T + 59 T^{2} + 206 T^{3} + 59 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 + 4 T + 41 T^{2} + 120 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 - 4 T + 21 T^{2} - 120 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 + 23 T^{2} + 128 T^{3} + 23 p T^{4} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 - 4 T + 61 T^{2} - 280 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - 10 T + 91 T^{2} - 748 T^{3} + 91 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 - 14 T + 151 T^{2} - 1026 T^{3} + 151 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 + 20 T + 241 T^{2} + 1838 T^{3} + 241 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 16 T + 221 T^{2} + 1628 T^{3} + 221 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 - 6 T + 59 T^{2} - 170 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 - 24 T + 353 T^{2} - 3232 T^{3} + 353 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 + 2 T + 179 T^{2} + 240 T^{3} + 179 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 - 10 T + 221 T^{2} - 1314 T^{3} + 221 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
71$C_2$ $$( 1 - 4 T + p T^{2} )^{3}$$
73$S_4\times C_2$ $$1 + 10 T + 231 T^{2} + 1420 T^{3} + 231 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 + 2 T + 37 T^{2} + 650 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 4 T + 161 T^{2} - 680 T^{3} + 161 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 - 12 T + 251 T^{2} - 1816 T^{3} + 251 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - 18 T + 255 T^{2} - 2412 T^{3} + 255 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$