# Properties

 Label 4-165564-1.1-c1e2-0-4 Degree $4$ Conductor $165564$ Sign $-1$ Analytic cond. $10.5565$ Root an. cond. $1.80251$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 4-s − 7-s − 2·13-s + 16-s − 4·19-s − 6·25-s − 28-s − 16·31-s + 6·43-s − 6·49-s − 2·52-s + 64-s − 3·73-s − 4·76-s − 4·79-s + 2·91-s − 16·97-s − 6·100-s + 4·103-s + 20·109-s − 112-s − 14·121-s − 16·124-s + 127-s + 131-s + 4·133-s + 137-s + ⋯
 L(s)  = 1 + 1/2·4-s − 0.377·7-s − 0.554·13-s + 1/4·16-s − 0.917·19-s − 6/5·25-s − 0.188·28-s − 2.87·31-s + 0.914·43-s − 6/7·49-s − 0.277·52-s + 1/8·64-s − 0.351·73-s − 0.458·76-s − 0.450·79-s + 0.209·91-s − 1.62·97-s − 3/5·100-s + 0.394·103-s + 1.91·109-s − 0.0944·112-s − 1.27·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$165564$$    =    $$2^{2} \cdot 3^{4} \cdot 7 \cdot 73$$ Sign: $-1$ Analytic conductor: $$10.5565$$ Root analytic conductor: $$1.80251$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{165564} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 165564,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
3 $$1$$
7$C_1$$\times$$C_2$ $$( 1 + T )( 1 + p T^{2} )$$
73$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 2 T + p T^{2} )$$
good5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
29$C_2^2$ $$1 - 26 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + p T^{2} )$$
47$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$C_2^2$ $$1 + 86 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 130 T^{2} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$