# Properties

 Label 4-3920e2-1.1-c0e2-0-5 Degree $4$ Conductor $15366400$ Sign $1$ Analytic cond. $3.82724$ Root an. cond. $1.39869$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3-s + 5-s + 9-s − 11-s + 2·13-s + 15-s − 17-s + 2·27-s − 2·29-s − 33-s + 2·39-s + 45-s + 47-s − 51-s − 55-s + 2·65-s − 4·71-s + 2·73-s − 79-s + 2·81-s + 4·83-s − 85-s − 2·87-s + 2·97-s − 99-s + 103-s + 109-s + ⋯
 L(s)  = 1 + 3-s + 5-s + 9-s − 11-s + 2·13-s + 15-s − 17-s + 2·27-s − 2·29-s − 33-s + 2·39-s + 45-s + 47-s − 51-s − 55-s + 2·65-s − 4·71-s + 2·73-s − 79-s + 2·81-s + 4·83-s − 85-s − 2·87-s + 2·97-s − 99-s + 103-s + 109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$15366400$$    =    $$2^{8} \cdot 5^{2} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$3.82724$$ Root analytic conductor: $$1.39869$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3920} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 15366400,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.828662815$$ $$L(\frac12)$$ $$\approx$$ $$2.828662815$$ $$L(1)$$ 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 $$1$$
5$C_2$ $$1 - T + T^{2}$$
7 $$1$$
good3$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
11$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
13$C_2$ $$( 1 - T + T^{2} )^{2}$$
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$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
29$C_2$ $$( 1 + T + T^{2} )^{2}$$
31$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
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_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
53$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
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$C_1$ $$( 1 + T )^{4}$$
73$C_2$ $$( 1 - T + T^{2} )^{2}$$
79$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
83$C_1$ $$( 1 - T )^{4}$$
89$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
97$C_2$ $$( 1 - T + T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$