# Properties

 Degree $4$ Conductor $4626801$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 4-s + 2·5-s + 2·8-s − 4·10-s + 2·11-s − 4·16-s + 2·17-s + 2·20-s − 4·22-s + 25-s + 2·29-s − 2·31-s + 2·32-s − 4·34-s + 4·40-s + 2·44-s − 2·50-s + 4·55-s − 4·58-s + 2·61-s + 4·62-s + 3·64-s − 2·67-s + 2·68-s − 8·80-s − 2·83-s + ⋯
 L(s)  = 1 − 2·2-s + 4-s + 2·5-s + 2·8-s − 4·10-s + 2·11-s − 4·16-s + 2·17-s + 2·20-s − 4·22-s + 25-s + 2·29-s − 2·31-s + 2·32-s − 4·34-s + 4·40-s + 2·44-s − 2·50-s + 4·55-s − 4·58-s + 2·61-s + 4·62-s + 3·64-s − 2·67-s + 2·68-s − 8·80-s − 2·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$4626801$$    =    $$3^{4} \cdot 239^{2}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{2151} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 4626801,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7209577906$$ $$L(\frac12)$$ $$\approx$$ $$0.7209577906$$ $$L(1)$$ 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)$
bad3 $$1$$
239$C_1$ $$( 1 - T )^{2}$$
good2$C_2$ $$( 1 + T + T^{2} )^{2}$$
5$C_2$ $$( 1 - T + T^{2} )^{2}$$
7$C_2^2$ $$1 + T^{4}$$
11$C_2$ $$( 1 - T + T^{2} )^{2}$$
13$C_2^2$ $$1 + T^{4}$$
17$C_2$ $$( 1 - T + T^{2} )^{2}$$
19$C_2^2$ $$1 + T^{4}$$
23$C_2^2$ $$1 + T^{4}$$
29$C_2$ $$( 1 - T + T^{2} )^{2}$$
31$C_2$ $$( 1 + T + T^{2} )^{2}$$
37$C_2^2$ $$1 + T^{4}$$
41$C_2^2$ $$1 + T^{4}$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
47$C_2^2$ $$1 + T^{4}$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
59$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
61$C_2$ $$( 1 - T + T^{2} )^{2}$$
67$C_2$ $$( 1 + T + T^{2} )^{2}$$
71$C_2$ $$( 1 + T^{2} )^{2}$$
73$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
79$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
83$C_2$ $$( 1 + T + T^{2} )^{2}$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
97$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$