# Properties

 Label 4-28e2-1.1-c1e2-0-0 Degree $4$ Conductor $784$ Sign $1$ Analytic cond. $0.0499885$ Root an. cond. $0.472843$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 8-s − 2·9-s − 2·10-s − 4·11-s − 2·12-s − 2·13-s − 4·15-s + 16-s + 2·18-s + 10·19-s + 2·20-s + 4·22-s + 2·24-s − 6·25-s + 2·26-s + 10·27-s + 4·30-s + 4·31-s − 32-s + 8·33-s − 2·36-s − 10·38-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.353·8-s − 2/3·9-s − 0.632·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s − 1.03·15-s + 1/4·16-s + 0.471·18-s + 2.29·19-s + 0.447·20-s + 0.852·22-s + 0.408·24-s − 6/5·25-s + 0.392·26-s + 1.92·27-s + 0.730·30-s + 0.718·31-s − 0.176·32-s + 1.39·33-s − 1/3·36-s − 1.62·38-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$784$$    =    $$2^{4} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$0.0499885$$ Root analytic conductor: $$0.472843$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 784,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.2887965445$$ $$L(\frac12)$$ $$\approx$$ $$0.2887965445$$ $$L(\frac{3}{2})$$ 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$C_1$ $$1 + T$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good3$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 2 T + p T^{2} )$$
5$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + p T^{2} )$$
11$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
43$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
53$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
59$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
71$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} )$$
83$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$