# Properties

 Label 4-117e2-1.1-c1e2-0-18 Degree $4$ Conductor $13689$ Sign $1$ Analytic cond. $0.872822$ Root an. cond. $0.966565$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s + 2·4-s − 4·5-s + 6·6-s − 2·7-s − 4·8-s + 6·9-s + 8·10-s − 2·11-s − 6·12-s − 13-s + 4·14-s + 12·15-s + 8·16-s − 10·17-s − 12·18-s − 12·19-s − 8·20-s + 6·21-s + 4·22-s + 3·23-s + 12·24-s + 5·25-s + 2·26-s − 9·27-s − 4·28-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s + 4-s − 1.78·5-s + 2.44·6-s − 0.755·7-s − 1.41·8-s + 2·9-s + 2.52·10-s − 0.603·11-s − 1.73·12-s − 0.277·13-s + 1.06·14-s + 3.09·15-s + 2·16-s − 2.42·17-s − 2.82·18-s − 2.75·19-s − 1.78·20-s + 1.30·21-s + 0.852·22-s + 0.625·23-s + 2.44·24-s + 25-s + 0.392·26-s − 1.73·27-s − 0.755·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$13689$$    =    $$3^{4} \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$0.872822$$ Root analytic conductor: $$0.966565$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 13689,\ (\ :1/2, 1/2),\ 1)$$

## 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)$
bad3$C_2$ $$1 + p T + p T^{2}$$
13$C_2$ $$1 + T + T^{2}$$
good2$C_2^2$ $$1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5$C_2^2$ $$1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
7$C_2^2$ $$1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
11$C_2^2$ $$1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
17$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
23$C_2^2$ $$1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
37$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
59$C_2^2$ $$1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
67$C_2^2$ $$1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
79$C_2^2$ $$1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
83$C_2^2$ $$1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
89$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
97$C_2^2$ $$1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$