# Properties

 Degree $4$ Conductor $36864$ Sign $1$ Motivic weight $2$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s − 12·7-s − 5·9-s − 20·13-s − 4·19-s + 24·21-s + 18·25-s + 28·27-s − 44·31-s + 12·37-s + 40·39-s − 164·43-s + 10·49-s + 8·57-s + 172·61-s + 60·63-s − 4·67-s + 164·73-s − 36·75-s + 20·79-s − 11·81-s + 240·91-s + 88·93-s − 188·97-s − 268·103-s − 20·109-s − 24·111-s + ⋯
 L(s)  = 1 − 2/3·3-s − 1.71·7-s − 5/9·9-s − 1.53·13-s − 0.210·19-s + 8/7·21-s + 0.719·25-s + 1.03·27-s − 1.41·31-s + 0.324·37-s + 1.02·39-s − 3.81·43-s + 0.204·49-s + 8/57·57-s + 2.81·61-s + 0.952·63-s − 0.0597·67-s + 2.24·73-s − 0.479·75-s + 0.253·79-s − 0.135·81-s + 2.63·91-s + 0.946·93-s − 1.93·97-s − 2.60·103-s − 0.183·109-s − 0.216·111-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$36864$$    =    $$2^{12} \cdot 3^{2}$$ Sign: $1$ Motivic weight: $$2$$ Character: induced by $\chi_{192} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 36864,\ (\ :1, 1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.370880$$ $$L(\frac12)$$ $$\approx$$ $$0.370880$$ $$L(2)$$ 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$C_2$ $$1 + 2 T + p^{2} T^{2}$$
good5$C_2^2$ $$1 - 18 T^{2} + p^{4} T^{4}$$
7$C_2$ $$( 1 + 6 T + p^{2} T^{2} )^{2}$$
11$C_2^2$ $$1 - 210 T^{2} + p^{4} T^{4}$$
13$C_2$ $$( 1 + 10 T + p^{2} T^{2} )^{2}$$
17$C_2^2$ $$1 - 66 T^{2} + p^{4} T^{4}$$
19$C_2$ $$( 1 + 2 T + p^{2} T^{2} )^{2}$$
23$C_2^2$ $$1 - 930 T^{2} + p^{4} T^{4}$$
29$C_2^2$ $$1 - 1394 T^{2} + p^{4} T^{4}$$
31$C_2$ $$( 1 + 22 T + p^{2} T^{2} )^{2}$$
37$C_2$ $$( 1 - 6 T + p^{2} T^{2} )^{2}$$
41$C_2^2$ $$1 - 2210 T^{2} + p^{4} T^{4}$$
43$C_2$ $$( 1 + 82 T + p^{2} T^{2} )^{2}$$
47$C_2^2$ $$1 + 190 T^{2} + p^{4} T^{4}$$
53$C_2^2$ $$1 - 1746 T^{2} + p^{4} T^{4}$$
59$C_2^2$ $$1 - 1554 T^{2} + p^{4} T^{4}$$
61$C_2$ $$( 1 - 86 T + p^{2} T^{2} )^{2}$$
67$C_2$ $$( 1 + 2 T + p^{2} T^{2} )^{2}$$
71$C_2^2$ $$1 + 5406 T^{2} + p^{4} T^{4}$$
73$C_2$ $$( 1 - 82 T + p^{2} T^{2} )^{2}$$
79$C_2$ $$( 1 - 10 T + p^{2} T^{2} )^{2}$$
83$C_2^2$ $$1 - 8370 T^{2} + p^{4} T^{4}$$
89$C_2^2$ $$1 - 14690 T^{2} + p^{4} T^{4}$$
97$C_2$ $$( 1 + 94 T + p^{2} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$