# Properties

 Label 4-192e2-1.1-c1e2-0-3 Degree $4$ Conductor $36864$ Sign $1$ Analytic cond. $2.35048$ Root an. cond. $1.23819$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2·7-s − 3·9-s + 4·11-s − 2·17-s − 4·19-s + 8·23-s + 2·25-s + 4·29-s + 10·31-s − 4·37-s − 2·41-s − 4·43-s + 8·47-s + 6·49-s − 4·53-s + 8·59-s − 4·61-s − 6·63-s − 8·67-s − 4·73-s + 8·77-s + 2·79-s + 9·81-s − 4·83-s + 2·89-s − 16·97-s − 12·99-s + ⋯
 L(s)  = 1 + 0.755·7-s − 9-s + 1.20·11-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 2/5·25-s + 0.742·29-s + 1.79·31-s − 0.657·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 6/7·49-s − 0.549·53-s + 1.04·59-s − 0.512·61-s − 0.755·63-s − 0.977·67-s − 0.468·73-s + 0.911·77-s + 0.225·79-s + 81-s − 0.439·83-s + 0.211·89-s − 1.62·97-s − 1.20·99-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$36864$$    =    $$2^{12} \cdot 3^{2}$$ Sign: $1$ Analytic conductor: $$2.35048$$ Root analytic conductor: $$1.23819$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 36864,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.392975217$$ $$L(\frac12)$$ $$\approx$$ $$1.392975217$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 + p T^{2}$$
good5$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
7$D_{4}$ $$1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
11$C_4$ $$1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + p T^{2} )$$
29$D_{4}$ $$1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
37$D_{4}$ $$1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$D_{4}$ $$1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
71$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
79$D_{4}$ $$1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$