# Properties

 Label 4-41472-1.1-c1e2-0-9 Degree $4$ Conductor $41472$ Sign $-1$ Analytic cond. $2.64429$ Root an. cond. $1.27519$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 5-s − 7-s − 3·9-s − 5·11-s − 3·13-s − 4·19-s − 3·23-s − 3·25-s + 29-s − 3·31-s − 35-s − 11·41-s + 15·43-s − 3·45-s − 47-s − 5·49-s + 8·53-s − 5·55-s + 5·59-s − 7·61-s + 3·63-s − 3·65-s + 9·67-s − 4·71-s + 8·73-s + 5·77-s + 3·79-s + ⋯
 L(s)  = 1 + 0.447·5-s − 0.377·7-s − 9-s − 1.50·11-s − 0.832·13-s − 0.917·19-s − 0.625·23-s − 3/5·25-s + 0.185·29-s − 0.538·31-s − 0.169·35-s − 1.71·41-s + 2.28·43-s − 0.447·45-s − 0.145·47-s − 5/7·49-s + 1.09·53-s − 0.674·55-s + 0.650·59-s − 0.896·61-s + 0.377·63-s − 0.372·65-s + 1.09·67-s − 0.474·71-s + 0.936·73-s + 0.569·77-s + 0.337·79-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 41472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 41472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$41472$$    =    $$2^{9} \cdot 3^{4}$$ Sign: $-1$ Analytic conductor: $$2.64429$$ Root analytic conductor: $$1.27519$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 41472,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad2 $$1$$
3$C_2$ $$1 + p T^{2}$$
good5$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$D_{4}$ $$1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 5 T + p T^{2} )$$
13$D_{4}$ $$1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$D_{4}$ $$1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
37$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 11 T + 64 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 - 15 T + 118 T^{2} - 15 p T^{3} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
53$D_{4}$ $$1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$D_{4}$ $$1 - 9 T + 118 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$D_{4}$ $$1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$D_{4}$ $$1 + 3 T - 56 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$