# Properties

 Label 4-1855e2-1.1-c1e2-0-6 Degree $4$ Conductor $3441025$ Sign $-1$ Analytic cond. $219.402$ Root an. cond. $3.84866$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 2·7-s − 5·9-s − 6·11-s + 10·13-s + 12·16-s + 6·17-s + 25-s − 8·28-s + 6·29-s + 20·36-s + 4·37-s − 20·43-s + 24·44-s + 18·47-s + 3·49-s − 40·52-s + 12·53-s − 10·63-s − 32·64-s − 24·68-s − 12·77-s + 16·81-s − 24·89-s + 20·91-s − 2·97-s + 30·99-s + ⋯
 L(s)  = 1 − 2·4-s + 0.755·7-s − 5/3·9-s − 1.80·11-s + 2.77·13-s + 3·16-s + 1.45·17-s + 1/5·25-s − 1.51·28-s + 1.11·29-s + 10/3·36-s + 0.657·37-s − 3.04·43-s + 3.61·44-s + 2.62·47-s + 3/7·49-s − 5.54·52-s + 1.64·53-s − 1.25·63-s − 4·64-s − 2.91·68-s − 1.36·77-s + 16/9·81-s − 2.54·89-s + 2.09·91-s − 0.203·97-s + 3.01·99-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3441025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3441025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$3441025$$    =    $$5^{2} \cdot 7^{2} \cdot 53^{2}$$ Sign: $-1$ Analytic conductor: $$219.402$$ Root analytic conductor: $$3.84866$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{3441025} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 3441025,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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)$
bad5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
7$C_1$ $$( 1 - T )^{2}$$
53$C_2$ $$1 - 12 T + p T^{2}$$
good2$C_2$ $$( 1 + p T^{2} )^{2}$$
3$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
11$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
29$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
43$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 9 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
79$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 + T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$