# Properties

 Label 4-21e4-1.1-c1e2-0-18 Degree $4$ Conductor $194481$ Sign $1$ Analytic cond. $12.4002$ Root an. cond. $1.87654$ 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 + 2·4-s + 5·8-s + 4·11-s + 5·16-s + 4·22-s + 8·23-s + 5·25-s − 4·29-s + 10·32-s + 6·37-s − 24·43-s + 8·44-s + 8·46-s + 5·50-s − 10·53-s − 4·58-s + 17·64-s − 4·67-s − 32·71-s + 6·74-s − 8·79-s − 24·86-s + 20·88-s + 16·92-s + 10·100-s − 10·106-s + ⋯
 L(s)  = 1 + 0.707·2-s + 4-s + 1.76·8-s + 1.20·11-s + 5/4·16-s + 0.852·22-s + 1.66·23-s + 25-s − 0.742·29-s + 1.76·32-s + 0.986·37-s − 3.65·43-s + 1.20·44-s + 1.17·46-s + 0.707·50-s − 1.37·53-s − 0.525·58-s + 17/8·64-s − 0.488·67-s − 3.79·71-s + 0.697·74-s − 0.900·79-s − 2.58·86-s + 2.13·88-s + 1.66·92-s + 100-s − 0.971·106-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$194481$$    =    $$3^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$12.4002$$ Root analytic conductor: $$1.87654$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{441} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 194481,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.737694151$$ $$L(\frac12)$$ $$\approx$$ $$3.737694151$$ $$L(\frac{3}{2})$$ 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)$
bad3 $$1$$
7 $$1$$
good2$C_2^2$ $$1 - T - T^{2} - p T^{3} + p^{2} T^{4}$$
5$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 + p T^{2} )^{2}$$
17$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
23$C_2^2$ $$1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
31$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4}$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
47$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
67$C_2^2$ $$1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
71$C_2$ $$( 1 + 16 T + p T^{2} )^{2}$$
73$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$