# Properties

 Label 4-642816-1.1-c1e2-0-5 Degree $4$ Conductor $642816$ Sign $-1$ Analytic cond. $40.9865$ Root an. cond. $2.53023$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 4·7-s + 2·13-s − 2·19-s − 8·25-s + 31-s + 16·37-s + 2·43-s + 2·49-s − 4·61-s + 2·67-s − 10·73-s + 10·79-s − 8·91-s + 12·97-s − 2·103-s + 8·109-s + 6·121-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
 L(s)  = 1 − 1.51·7-s + 0.554·13-s − 0.458·19-s − 8/5·25-s + 0.179·31-s + 2.63·37-s + 0.304·43-s + 2/7·49-s − 0.512·61-s + 0.244·67-s − 1.17·73-s + 1.12·79-s − 0.838·91-s + 1.21·97-s − 0.197·103-s + 0.766·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$642816$$    =    $$2^{8} \cdot 3^{4} \cdot 31$$ Sign: $-1$ Analytic conductor: $$40.9865$$ Root analytic conductor: $$2.53023$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{642816} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 642816,\ (\ :1/2, 1/2),\ -1)$$

## 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 $$1$$
31$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 2 T + p T^{2} )$$
good5$C_2^2$ $$1 + 8 T^{2} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + p T^{2} )$$
17$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 34 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} )$$
41$C_2^2$ $$1 + 8 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2^2$ $$1 + 86 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 74 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + p T^{2} )$$
71$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$