# Properties

 Label 4-3332e2-1.1-c0e2-0-4 Degree $4$ Conductor $11102224$ Sign $1$ Analytic cond. $2.76518$ Root an. cond. $1.28952$ 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 − 3-s + 6-s + 8-s + 9-s + 11-s + 2·13-s − 16-s + 17-s − 18-s − 22-s − 2·23-s − 24-s − 25-s − 2·26-s − 2·27-s + 2·31-s − 33-s − 34-s − 2·39-s + 2·46-s + 48-s + 50-s − 51-s + 53-s + 2·54-s − 2·62-s + ⋯
 L(s)  = 1 − 2-s − 3-s + 6-s + 8-s + 9-s + 11-s + 2·13-s − 16-s + 17-s − 18-s − 22-s − 2·23-s − 24-s − 25-s − 2·26-s − 2·27-s + 2·31-s − 33-s − 34-s − 2·39-s + 2·46-s + 48-s + 50-s − 51-s + 53-s + 2·54-s − 2·62-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$11102224$$    =    $$2^{4} \cdot 7^{4} \cdot 17^{2}$$ Sign: $1$ Analytic conductor: $$2.76518$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 11102224,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6443010657$$ $$L(\frac12)$$ $$\approx$$ $$0.6443010657$$ $$L(1)$$ 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$C_2$ $$1 + T + T^{2}$$
7 $$1$$
17$C_2$ $$1 - T + T^{2}$$
good3$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
5$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
11$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
13$C_2$ $$( 1 - T + T^{2} )^{2}$$
19$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
23$C_2$ $$( 1 + T + T^{2} )^{2}$$
29$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
31$C_2$ $$( 1 - T + T^{2} )^{2}$$
37$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
41$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
47$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
53$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
59$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
61$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
67$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
71$C_2$ $$( 1 + T + T^{2} )^{2}$$
73$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
79$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
83$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
89$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
97$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$