# Properties

 Label 4-148e2-1.1-c0e2-0-0 Degree $4$ Conductor $21904$ Sign $1$ Analytic cond. $0.00545553$ Root an. cond. $0.271774$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 5-s + 8-s − 9-s − 10-s − 2·13-s − 16-s + 17-s + 18-s + 25-s + 2·26-s − 2·29-s − 34-s − 37-s + 40-s + 41-s − 45-s − 49-s − 50-s − 2·53-s + 2·58-s + 61-s + 64-s − 2·65-s − 72-s + 4·73-s + 74-s + ⋯
 L(s)  = 1 − 2-s + 5-s + 8-s − 9-s − 10-s − 2·13-s − 16-s + 17-s + 18-s + 25-s + 2·26-s − 2·29-s − 34-s − 37-s + 40-s + 41-s − 45-s − 49-s − 50-s − 2·53-s + 2·58-s + 61-s + 64-s − 2·65-s − 72-s + 4·73-s + 74-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$21904$$    =    $$2^{4} \cdot 37^{2}$$ Sign: $1$ Analytic conductor: $$0.00545553$$ Root analytic conductor: $$0.271774$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{148} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 21904,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.2597781081$$ $$L(\frac12)$$ $$\approx$$ $$0.2597781081$$ $$L(1)$$ 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$C_2$ $$1 + T + T^{2}$$
37$C_2$ $$1 + T + T^{2}$$
good3$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
5$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
7$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
11$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
13$C_2$ $$( 1 + T + T^{2} )^{2}$$
17$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
19$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
23$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
29$C_2$ $$( 1 + T + T^{2} )^{2}$$
31$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
41$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
47$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
53$C_2$ $$( 1 + T + T^{2} )^{2}$$
59$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
61$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
67$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
71$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
73$C_1$ $$( 1 - T )^{4}$$
79$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
83$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
89$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
97$C_2$ $$( 1 + T + T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$