# Properties

 Label 4-1494108-1.1-c1e2-0-0 Degree $4$ Conductor $1494108$ Sign $1$ Analytic cond. $95.2656$ Root an. cond. $3.12416$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 3·4-s − 7-s − 4·8-s + 9-s − 2·11-s + 2·14-s + 5·16-s − 2·18-s + 4·22-s − 8·23-s − 10·25-s − 3·28-s + 12·29-s − 6·32-s + 3·36-s + 20·37-s + 16·43-s − 6·44-s + 16·46-s + 49-s + 20·50-s − 20·53-s + 4·56-s − 24·58-s − 63-s + 7·64-s + ⋯
 L(s)  = 1 − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 1/3·9-s − 0.603·11-s + 0.534·14-s + 5/4·16-s − 0.471·18-s + 0.852·22-s − 1.66·23-s − 2·25-s − 0.566·28-s + 2.22·29-s − 1.06·32-s + 1/2·36-s + 3.28·37-s + 2.43·43-s − 0.904·44-s + 2.35·46-s + 1/7·49-s + 2.82·50-s − 2.74·53-s + 0.534·56-s − 3.15·58-s − 0.125·63-s + 7/8·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1494108$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$95.2656$$ Root analytic conductor: $$3.12416$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{1494108} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1494108,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7197813021$$ $$L(\frac12)$$ $$\approx$$ $$0.7197813021$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 + T )^{2}$$
3$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
7$C_1$ $$1 + T$$
11$C_1$ $$( 1 + T )^{2}$$
good5$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
19$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
37$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
43$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
53$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 - 16 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
79$C_2$ $$( 1 + 16 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$