# Properties

 Label 4-192e2-1.1-c6e2-0-0 Degree $4$ Conductor $36864$ Sign $1$ Analytic cond. $1951.02$ Root an. cond. $6.64608$ Motivic weight $6$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 300·5-s − 243·9-s − 6.78e3·13-s + 1.03e4·17-s + 3.62e4·25-s − 6.42e4·29-s + 1.52e5·37-s − 1.40e5·41-s + 7.29e4·45-s + 1.29e5·49-s − 1.33e5·53-s + 5.14e5·61-s + 2.03e6·65-s + 4.86e5·73-s + 5.90e4·81-s − 3.10e6·85-s − 1.37e6·89-s − 1.88e6·97-s + 5.19e5·101-s + 2.04e6·109-s − 2.62e6·113-s + 1.64e6·117-s + 1.36e6·121-s + 5.62e5·125-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 2.39·5-s − 1/3·9-s − 3.08·13-s + 2.10·17-s + 2.31·25-s − 2.63·29-s + 3.00·37-s − 2.03·41-s + 4/5·45-s + 1.09·49-s − 0.899·53-s + 2.26·61-s + 7.41·65-s + 1.25·73-s + 1/9·81-s − 5.05·85-s − 1.94·89-s − 2.06·97-s + 0.504·101-s + 1.58·109-s − 1.82·113-s + 1.02·117-s + 0.770·121-s + 0.287·125-s + 6.32·145-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$36864$$    =    $$2^{12} \cdot 3^{2}$$ Sign: $1$ Analytic conductor: $$1951.02$$ Root analytic conductor: $$6.64608$$ Motivic weight: $$6$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 36864,\ (\ :3, 3),\ 1)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$9.563712700\times10^{-6}$$ $$L(\frac12)$$ $$\approx$$ $$9.563712700\times10^{-6}$$ $$L(4)$$ 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$C_2$ $$1 + p^{5} T^{2}$$
good5$C_2$ $$( 1 + 6 p^{2} T + p^{6} T^{2} )^{2}$$
7$C_2^2$ $$1 - 129266 T^{2} + p^{12} T^{4}$$
11$C_2^2$ $$1 - 1365410 T^{2} + p^{12} T^{4}$$
13$C_2$ $$( 1 + 3394 T + p^{6} T^{2} )^{2}$$
17$C_2$ $$( 1 - 5178 T + p^{6} T^{2} )^{2}$$
19$C_2^2$ $$1 - 47709890 T^{2} + p^{12} T^{4}$$
23$C_2^2$ $$1 - 280146530 T^{2} + p^{12} T^{4}$$
29$C_2$ $$( 1 + 32142 T + p^{6} T^{2} )^{2}$$
31$C_2^2$ $$1 - 711526610 T^{2} + p^{12} T^{4}$$
37$C_2$ $$( 1 - 76150 T + p^{6} T^{2} )^{2}$$
41$C_2$ $$( 1 + 70038 T + p^{6} T^{2} )^{2}$$
43$C_2^2$ $$1 - 2451652130 T^{2} + p^{12} T^{4}$$
47$C_2^2$ $$1 + 1425021694 T^{2} + p^{12} T^{4}$$
53$C_2$ $$( 1 + 66942 T + p^{6} T^{2} )^{2}$$
59$C_2^2$ $$1 + 68178858910 T^{2} + p^{12} T^{4}$$
61$C_2$ $$( 1 - 257014 T + p^{6} T^{2} )^{2}$$
67$C_2^2$ $$1 - 77748332930 T^{2} + p^{12} T^{4}$$
71$C_2^2$ $$1 - 138474492194 T^{2} + p^{12} T^{4}$$
73$C_2$ $$( 1 - 243442 T + p^{6} T^{2} )^{2}$$
79$C_2^2$ $$1 - 260585085842 T^{2} + p^{12} T^{4}$$
83$C_2^2$ $$1 + 415389237694 T^{2} + p^{12} T^{4}$$
89$C_2$ $$( 1 + 686766 T + p^{6} T^{2} )^{2}$$
97$C_2$ $$( 1 + 942686 T + p^{6} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$