# Properties

 Label 4-112e3-1.1-c1e2-0-11 Degree $4$ Conductor $1404928$ Sign $1$ Analytic cond. $89.5794$ Root an. cond. $3.07646$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 7-s + 4·9-s + 4·11-s + 4·23-s − 8·25-s + 6·29-s + 2·37-s + 4·43-s + 49-s − 18·53-s + 4·63-s + 4·77-s + 7·81-s + 16·99-s − 16·107-s + 2·109-s − 8·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + 163-s + ⋯
 L(s)  = 1 + 0.377·7-s + 4/3·9-s + 1.20·11-s + 0.834·23-s − 8/5·25-s + 1.11·29-s + 0.328·37-s + 0.609·43-s + 1/7·49-s − 2.47·53-s + 0.503·63-s + 0.455·77-s + 7/9·81-s + 1.60·99-s − 1.54·107-s + 0.191·109-s − 0.752·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.315·161-s + 0.0783·163-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1404928$$    =    $$2^{12} \cdot 7^{3}$$ Sign: $1$ Analytic conductor: $$89.5794$$ Root analytic conductor: $$3.07646$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1404928,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.922655939$$ $$L(\frac12)$$ $$\approx$$ $$2.922655939$$ $$L(\frac{3}{2})$$ 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$C_1$ $$1 - T$$
good3$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
5$C_2^2$ $$1 + 8 T^{2} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
13$C_2^2$ $$1 - 16 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + p T^{2} )$$
31$C_2^2$ $$1 + 50 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + p T^{2} )$$
41$C_2^2$ $$1 + 50 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
47$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
59$C_2^2$ $$1 - 52 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 72 T^{2} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
71$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
79$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2^2$ $$1 + 44 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
97$C_2^2$ $$1 + 66 T^{2} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$