# Properties

 Label 4-192e2-1.1-c1e2-0-7 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 + 2·3-s + 9-s + 4·11-s + 4·13-s − 8·23-s − 2·25-s − 4·27-s + 8·33-s + 4·37-s + 8·39-s − 2·49-s − 4·59-s + 20·61-s − 16·69-s − 8·71-s + 4·73-s − 4·75-s − 11·81-s − 20·83-s − 4·97-s + 4·99-s − 4·107-s − 12·109-s + 8·111-s + 4·117-s + 6·121-s + 127-s + ⋯
 L(s)  = 1 + 1.15·3-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 1.66·23-s − 2/5·25-s − 0.769·27-s + 1.39·33-s + 0.657·37-s + 1.28·39-s − 2/7·49-s − 0.520·59-s + 2.56·61-s − 1.92·69-s − 0.949·71-s + 0.468·73-s − 0.461·75-s − 1.22·81-s − 2.19·83-s − 0.406·97-s + 0.402·99-s − 0.386·107-s − 1.14·109-s + 0.759·111-s + 0.369·117-s + 6/11·121-s + 0.0887·127-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

 $$L(1)$$ $$\approx$$ $$1.964811619$$ $$L(\frac12)$$ $$\approx$$ $$1.964811619$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 - 2 T + p T^{2}$$
good5$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
7$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
23$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
29$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
41$C_2^2$ $$1 + 30 T^{2} + p^{2} T^{4}$$
43$C_2^2$ $$1 + 66 T^{2} + p^{2} T^{4}$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} )$$
67$C_2^2$ $$1 - 94 T^{2} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$C_2^2$ $$1 - 94 T^{2} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
89$C_2^2$ $$1 - 82 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$