# Properties

 Label 4-131e2-1.1-c0e2-0-0 Degree $4$ Conductor $17161$ Sign $1$ Analytic cond. $0.00427421$ Root an. cond. $0.255690$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3-s + 2·4-s − 5-s − 7-s − 11-s − 2·12-s − 13-s + 15-s + 3·16-s − 2·20-s + 21-s − 2·28-s + 33-s + 35-s + 39-s − 41-s − 43-s − 2·44-s − 3·48-s − 2·52-s + 4·53-s + 55-s − 59-s + 2·60-s − 61-s + 4·64-s + 65-s + ⋯
 L(s)  = 1 − 3-s + 2·4-s − 5-s − 7-s − 11-s − 2·12-s − 13-s + 15-s + 3·16-s − 2·20-s + 21-s − 2·28-s + 33-s + 35-s + 39-s − 41-s − 43-s − 2·44-s − 3·48-s − 2·52-s + 4·53-s + 55-s − 59-s + 2·60-s − 61-s + 4·64-s + 65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 17161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 17161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$17161$$    =    $$131^{2}$$ Sign: $1$ Analytic conductor: $$0.00427421$$ Root analytic conductor: $$0.255690$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{131} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 17161,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3263700811$$ $$L(\frac12)$$ $$\approx$$ $$0.3263700811$$ $$L(1)$$ 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)$
bad131$C_1$ $$( 1 - T )^{2}$$
good2$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
3$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
5$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
7$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
11$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
13$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
17$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
23$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
29$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
31$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
37$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
41$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
43$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
47$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
53$C_1$ $$( 1 - T )^{4}$$
59$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
61$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
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_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
89$C_1$ $$( 1 - T )^{4}$$
97$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$