# Properties

 Label 6-7800e3-1.1-c1e3-0-9 Degree $6$ Conductor $474552000000$ Sign $-1$ Analytic cond. $241610.$ Root an. cond. $7.89197$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·3-s − 5·7-s + 6·9-s + 3·11-s − 3·13-s − 17-s + 4·19-s + 15·21-s − 5·23-s − 10·27-s − 4·29-s + 10·31-s − 9·33-s − 5·37-s + 9·39-s + 5·41-s − 12·43-s − 2·47-s + 10·49-s + 3·51-s + 13·53-s − 12·57-s + 21·61-s − 30·63-s − 8·67-s + 15·69-s − 71-s + ⋯
 L(s)  = 1 − 1.73·3-s − 1.88·7-s + 2·9-s + 0.904·11-s − 0.832·13-s − 0.242·17-s + 0.917·19-s + 3.27·21-s − 1.04·23-s − 1.92·27-s − 0.742·29-s + 1.79·31-s − 1.56·33-s − 0.821·37-s + 1.44·39-s + 0.780·41-s − 1.82·43-s − 0.291·47-s + 10/7·49-s + 0.420·51-s + 1.78·53-s − 1.58·57-s + 2.68·61-s − 3.77·63-s − 0.977·67-s + 1.80·69-s − 0.118·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}$$ Sign: $-1$ Analytic conductor: $$241610.$$ Root analytic conductor: $$7.89197$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 + T )^{3}$$
5 $$1$$
13$C_1$ $$( 1 + T )^{3}$$
good7$A_4\times C_2$ $$1 + 5 T + 15 T^{2} + 38 T^{3} + 15 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
11$C_6$ $$1 - 3 T - 7 T^{2} + 62 T^{3} - 7 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
17$A_4\times C_2$ $$1 + T + 37 T^{2} + 42 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
19$A_4\times C_2$ $$1 - 4 T + 5 T^{2} + 24 T^{3} + 5 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
23$A_4\times C_2$ $$1 + 5 T + 63 T^{2} + 198 T^{3} + 63 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
29$A_4\times C_2$ $$1 + 4 T + 35 T^{2} + 56 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
31$A_4\times C_2$ $$1 - 10 T + 69 T^{2} - 364 T^{3} + 69 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
37$A_4\times C_2$ $$1 + 5 T + 19 T^{2} - 178 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
41$A_4\times C_2$ $$1 - 5 T + 31 T^{2} + 138 T^{3} + 31 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{3}$$
47$A_4\times C_2$ $$1 + 2 T + 85 T^{2} + 252 T^{3} + 85 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
53$A_4\times C_2$ $$1 - 13 T + 115 T^{2} - 894 T^{3} + 115 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6}$$
59$C_2$ $$( 1 + p T^{2} )^{3}$$
61$A_4\times C_2$ $$1 - 21 T + 287 T^{2} - 2518 T^{3} + 287 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6}$$
67$A_4\times C_2$ $$1 + 8 T - 7 T^{2} - 336 T^{3} - 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
71$A_4\times C_2$ $$1 + T + 113 T^{2} + 494 T^{3} + 113 p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
73$A_4\times C_2$ $$1 + 28 T + 423 T^{2} + 4264 T^{3} + 423 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6}$$
79$A_4\times C_2$ $$1 + 21 T + 341 T^{2} + 3446 T^{3} + 341 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6}$$
83$A_4\times C_2$ $$1 + 4 T + 25 T^{2} + 1176 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
89$A_4\times C_2$ $$1 - 11 T + 35 T^{2} + 638 T^{3} + 35 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}$$
97$A_4\times C_2$ $$1 + 25 T + 5 p T^{2} + 5322 T^{3} + 5 p^{2} T^{4} + 25 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$