# Properties

 Label 4-384e2-1.1-c1e2-0-29 Degree $4$ Conductor $147456$ Sign $-1$ Analytic cond. $9.40192$ Root an. cond. $1.75107$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 4·5-s + 9-s + 4·17-s + 6·25-s − 12·29-s + 8·37-s − 12·41-s − 4·45-s + 6·49-s + 4·53-s − 8·61-s + 12·73-s + 81-s − 16·85-s − 12·89-s − 12·97-s + 20·101-s − 16·109-s − 12·113-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + ⋯
 L(s)  = 1 − 1.78·5-s + 1/3·9-s + 0.970·17-s + 6/5·25-s − 2.22·29-s + 1.31·37-s − 1.87·41-s − 0.596·45-s + 6/7·49-s + 0.549·53-s − 1.02·61-s + 1.40·73-s + 1/9·81-s − 1.73·85-s − 1.27·89-s − 1.21·97-s + 1.99·101-s − 1.53·109-s − 1.12·113-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$147456$$    =    $$2^{14} \cdot 3^{2}$$ Sign: $-1$ Analytic conductor: $$9.40192$$ Root analytic conductor: $$1.75107$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{147456} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 147456,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good5$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
31$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
59$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2^2$ $$1 - 106 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
79$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2^2$ $$1 - 42 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$