# Properties

 Label 6-4275e3-1.1-c1e3-0-4 Degree $6$ Conductor $78128296875$ Sign $1$ Analytic cond. $39777.7$ Root an. cond. $5.84260$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 7·4-s + 7·8-s − 11-s − 5·13-s + 7·16-s + 2·17-s − 3·19-s − 4·22-s + 8·23-s − 20·26-s + 7·29-s − 5·31-s + 14·32-s + 8·34-s − 5·37-s − 12·38-s − 41-s − 5·43-s − 7·44-s + 32·46-s + 3·47-s − 14·49-s − 35·52-s + 19·53-s + 28·58-s − 10·59-s + ⋯
 L(s)  = 1 + 2.82·2-s + 7/2·4-s + 2.47·8-s − 0.301·11-s − 1.38·13-s + 7/4·16-s + 0.485·17-s − 0.688·19-s − 0.852·22-s + 1.66·23-s − 3.92·26-s + 1.29·29-s − 0.898·31-s + 2.47·32-s + 1.37·34-s − 0.821·37-s − 1.94·38-s − 0.156·41-s − 0.762·43-s − 1.05·44-s + 4.71·46-s + 0.437·47-s − 2·49-s − 4.85·52-s + 2.60·53-s + 3.67·58-s − 1.30·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$3^{6} \cdot 5^{6} \cdot 19^{3}$$ Sign: $1$ Analytic conductor: $$39777.7$$ Root analytic conductor: $$5.84260$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$18.07076922$$ $$L(\frac12)$$ $$\approx$$ $$18.07076922$$ $$L(\frac{3}{2})$$ 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)$
bad3 $$1$$
5 $$1$$
19$C_1$ $$( 1 + T )^{3}$$
good2$C_6$ $$1 - p^{2} T + 9 T^{2} - 15 T^{3} + 9 p T^{4} - p^{4} T^{5} + p^{3} T^{6}$$
7$A_4\times C_2$ $$1 + 2 p T^{2} - p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6}$$
11$A_4\times C_2$ $$1 + T + 17 T^{2} + 35 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
13$A_4\times C_2$ $$1 + 5 T + 45 T^{2} + 131 T^{3} + 45 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
17$A_4\times C_2$ $$1 - 2 T + 36 T^{2} - 81 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
23$A_4\times C_2$ $$1 - 8 T + 88 T^{2} - 381 T^{3} + 88 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
29$A_4\times C_2$ $$1 - 7 T + 45 T^{2} - 315 T^{3} + 45 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}$$
31$A_4\times C_2$ $$1 + 5 T + 57 T^{2} + 353 T^{3} + 57 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
37$A_4\times C_2$ $$1 + 5 T + 103 T^{2} + 371 T^{3} + 103 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
41$A_4\times C_2$ $$1 + T + 9 T^{2} - 339 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
43$A_4\times C_2$ $$1 + 5 T + 72 T^{2} + 137 T^{3} + 72 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
47$A_4\times C_2$ $$1 - 3 T + 137 T^{2} - 269 T^{3} + 137 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
53$A_4\times C_2$ $$1 - 19 T + 214 T^{2} - 1707 T^{3} + 214 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6}$$
59$A_4\times C_2$ $$1 + 10 T + 145 T^{2} + 852 T^{3} + 145 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
61$A_4\times C_2$ $$1 + 17 T + 249 T^{2} + 2033 T^{3} + 249 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6}$$
67$A_4\times C_2$ $$1 - T + 59 T^{2} - 693 T^{3} + 59 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
71$A_4\times C_2$ $$1 - 19 T + 268 T^{2} - 2391 T^{3} + 268 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6}$$
73$A_4\times C_2$ $$1 - T + 49 T^{2} + 317 T^{3} + 49 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
79$A_4\times C_2$ $$1 - 18 T + 324 T^{2} - 2941 T^{3} + 324 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}$$
83$A_4\times C_2$ $$1 - 13 T + 247 T^{2} - 2019 T^{3} + 247 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6}$$
89$A_4\times C_2$ $$1 + 2 T + 168 T^{2} + 75 T^{3} + 168 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
97$A_4\times C_2$ $$1 + 5 T + 297 T^{2} + 971 T^{3} + 297 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$