# Properties

 Label 4-728e2-1.1-c1e2-0-16 Degree $4$ Conductor $529984$ Sign $1$ Analytic cond. $33.7922$ Root an. cond. $2.41103$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2·3-s + 4·7-s + 9-s − 12·19-s + 8·21-s + 6·25-s + 2·27-s + 4·29-s − 12·31-s + 16·37-s + 20·47-s + 9·49-s + 4·53-s − 24·57-s + 8·59-s + 4·63-s + 12·75-s + 4·81-s + 12·83-s + 8·87-s − 24·93-s − 12·103-s − 8·109-s + 32·111-s + 14·113-s − 15·121-s + 127-s + ⋯
 L(s)  = 1 + 1.15·3-s + 1.51·7-s + 1/3·9-s − 2.75·19-s + 1.74·21-s + 6/5·25-s + 0.384·27-s + 0.742·29-s − 2.15·31-s + 2.63·37-s + 2.91·47-s + 9/7·49-s + 0.549·53-s − 3.17·57-s + 1.04·59-s + 0.503·63-s + 1.38·75-s + 4/9·81-s + 1.31·83-s + 0.857·87-s − 2.48·93-s − 1.18·103-s − 0.766·109-s + 3.03·111-s + 1.31·113-s − 1.36·121-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$529984$$    =    $$2^{6} \cdot 7^{2} \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$33.7922$$ Root analytic conductor: $$2.41103$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 529984,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.425094978$$ $$L(\frac12)$$ $$\approx$$ $$3.425094978$$ $$L(\frac{3}{2})$$ 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 $$1$$
7$C_2$ $$1 - 4 T + p T^{2}$$
13$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good3$C_2$$\times$$C_2$ $$( 1 - p T + p T^{2} )( 1 + T + p T^{2} )$$
5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2^2$ $$1 + 15 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
23$C_2^2$ $$1 - 15 T^{2} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} )$$
41$C_2^2$ $$1 + 63 T^{2} + p^{2} T^{4}$$
43$C_2^2$ $$1 + 38 T^{2} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 - 9 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + p T^{2} )$$
61$C_2^2$ $$1 - 31 T^{2} + p^{2} T^{4}$$
67$C_2^2$ $$1 + 71 T^{2} + p^{2} T^{4}$$
71$C_2^2$ $$1 + 18 T^{2} + p^{2} T^{4}$$
73$C_2^2$ $$1 + 55 T^{2} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 25 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
89$C_2^2$ $$1 - 110 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 - 41 T^{2} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$