# Properties

 Label 4-8788-1.1-c1e2-0-0 Degree $4$ Conductor $8788$ Sign $-1$ Analytic cond. $0.560330$ Root an. cond. $0.865189$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·3-s + 4-s + 21·9-s − 6·12-s − 13-s + 16-s − 6·17-s − 8·23-s − 9·25-s − 54·27-s + 4·29-s + 21·36-s + 6·39-s − 10·43-s − 6·48-s − 13·49-s + 36·51-s − 52-s + 24·53-s − 16·61-s + 64-s − 6·68-s + 48·69-s + 54·75-s − 8·79-s + 108·81-s − 24·87-s + ⋯
 L(s)  = 1 − 3.46·3-s + 1/2·4-s + 7·9-s − 1.73·12-s − 0.277·13-s + 1/4·16-s − 1.45·17-s − 1.66·23-s − 9/5·25-s − 10.3·27-s + 0.742·29-s + 7/2·36-s + 0.960·39-s − 1.52·43-s − 0.866·48-s − 1.85·49-s + 5.04·51-s − 0.138·52-s + 3.29·53-s − 2.04·61-s + 1/8·64-s − 0.727·68-s + 5.77·69-s + 6.23·75-s − 0.900·79-s + 12·81-s − 2.57·87-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$8788$$    =    $$2^{2} \cdot 13^{3}$$ Sign: $-1$ Analytic conductor: $$0.560330$$ Root analytic conductor: $$0.865189$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 8788,\ (\ :1/2, 1/2),\ -1)$$

## 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$ $$( 1 - T )( 1 + T )$$
13$C_1$ $$1 + T$$
good3$C_2$ $$( 1 + p T + p T^{2} )^{2}$$
5$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
7$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
11$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
53$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
61$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
71$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$