# Properties

 Label 4-4752-1.1-c1e2-0-5 Degree $4$ Conductor $4752$ Sign $1$ Analytic cond. $0.302991$ Root an. cond. $0.741920$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s − 3·11-s − 2·12-s − 2·13-s − 4·16-s + 2·18-s − 6·22-s + 2·23-s + 6·25-s − 4·26-s − 27-s − 8·32-s + 3·33-s + 2·36-s − 4·37-s + 2·39-s − 6·44-s + 4·46-s + 4·47-s + 4·48-s − 10·49-s + 12·50-s − 4·52-s + ⋯
 L(s)  = 1 + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 0.554·13-s − 16-s + 0.471·18-s − 1.27·22-s + 0.417·23-s + 6/5·25-s − 0.784·26-s − 0.192·27-s − 1.41·32-s + 0.522·33-s + 1/3·36-s − 0.657·37-s + 0.320·39-s − 0.904·44-s + 0.589·46-s + 0.583·47-s + 0.577·48-s − 1.42·49-s + 1.69·50-s − 0.554·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$4752$$    =    $$2^{4} \cdot 3^{3} \cdot 11$$ Sign: $1$ Analytic conductor: $$0.302991$$ Root analytic conductor: $$0.741920$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 4752,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.268349092$$ $$L(\frac12)$$ $$\approx$$ $$1.268349092$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 - p T + p T^{2}$$
3$C_1$ $$1 + T$$
11$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 2 T + p T^{2} )$$
good5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2^2$ $$1 + 20 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
29$C_2^2$ $$1 + 8 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 + 18 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 - 32 T^{2} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + p T^{2} )$$
61$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
73$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$C_2^2$ $$1 + 18 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
89$C_2^2$ $$1 - 42 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$