# Properties

 Label 6-370e3-1.1-c1e3-0-0 Degree $6$ Conductor $50653000$ Sign $1$ Analytic cond. $25.7891$ Root an. cond. $1.71885$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s + 6·4-s − 3·5-s − 7-s + 10·8-s + 9-s − 9·10-s + 11·11-s − 3·14-s + 15·16-s + 17-s + 3·18-s − 18·20-s + 33·22-s − 2·23-s + 6·25-s − 4·27-s − 6·28-s − 5·29-s + 3·31-s + 21·32-s + 3·34-s + 3·35-s + 6·36-s − 3·37-s − 30·40-s − 9·41-s + ⋯
 L(s)  = 1 + 2.12·2-s + 3·4-s − 1.34·5-s − 0.377·7-s + 3.53·8-s + 1/3·9-s − 2.84·10-s + 3.31·11-s − 0.801·14-s + 15/4·16-s + 0.242·17-s + 0.707·18-s − 4.02·20-s + 7.03·22-s − 0.417·23-s + 6/5·25-s − 0.769·27-s − 1.13·28-s − 0.928·29-s + 0.538·31-s + 3.71·32-s + 0.514·34-s + 0.507·35-s + 36-s − 0.493·37-s − 4.74·40-s − 1.40·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 50653000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 50653000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$50653000$$    =    $$2^{3} \cdot 5^{3} \cdot 37^{3}$$ Sign: $1$ Analytic conductor: $$25.7891$$ Root analytic conductor: $$1.71885$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 50653000,\ (\ :1/2, 1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$7.126132790$$ $$L(\frac12)$$ $$\approx$$ $$7.126132790$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 - T )^{3}$$
5$C_1$ $$( 1 + T )^{3}$$
37$C_1$ $$( 1 + T )^{3}$$
good3$S_4\times C_2$ $$1 - T^{2} + 4 T^{3} - p T^{4} + p^{3} T^{6}$$
7$S_4\times C_2$ $$1 + T + 13 T^{2} + 4 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 - p T + 61 T^{2} - 234 T^{3} + 61 p T^{4} - p^{3} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 - T^{2} - 32 T^{3} - p T^{4} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 - T + 39 T^{2} - 42 T^{3} + 39 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 + 47 T^{2} + 4 T^{3} + 47 p T^{4} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 + 2 T + 21 T^{2} + 156 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 + 5 T + 71 T^{2} + 214 T^{3} + 71 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 - 3 T + 21 T^{2} + 84 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 + 9 T + 83 T^{2} + 374 T^{3} + 83 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 + 11 T + 145 T^{2} + 866 T^{3} + 145 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 - 2 T + 111 T^{2} - 132 T^{3} + 111 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 - 21 T + 239 T^{2} - 1910 T^{3} + 239 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 - 14 T + 211 T^{2} - 1572 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 7 T + 175 T^{2} - 850 T^{3} + 175 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 - 2 T + 171 T^{2} - 212 T^{3} + 171 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 - 22 T + 277 T^{2} - 2484 T^{3} + 277 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 16 T + 255 T^{2} + 2128 T^{3} + 255 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 + 26 T + 407 T^{2} + 4332 T^{3} + 407 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 2 T + 91 T^{2} - 12 p T^{3} + 91 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )^{3}$$
97$S_4\times C_2$ $$1 + 21 T + 371 T^{2} + 3758 T^{3} + 371 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$