# Properties

 Degree $4$ Conductor $31329$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 2·3-s − 2·4-s + 2·6-s − 7·7-s + 3·8-s + 3·9-s + 4·12-s − 8·13-s + 7·14-s + 16-s − 3·17-s − 3·18-s − 5·19-s + 14·21-s − 7·23-s − 6·24-s − 5·25-s + 8·26-s − 4·27-s + 14·28-s + 15·29-s − 31-s − 2·32-s + 3·34-s − 6·36-s − 7·37-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s − 2.64·7-s + 1.06·8-s + 9-s + 1.15·12-s − 2.21·13-s + 1.87·14-s + 1/4·16-s − 0.727·17-s − 0.707·18-s − 1.14·19-s + 3.05·21-s − 1.45·23-s − 1.22·24-s − 25-s + 1.56·26-s − 0.769·27-s + 2.64·28-s + 2.78·29-s − 0.179·31-s − 0.353·32-s + 0.514·34-s − 36-s − 1.15·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 31329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 31329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$31329$$    =    $$3^{2} \cdot 59^{2}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{177} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 31329,\ (\ :1/2, 1/2),\ 1)$$

## 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$C_1$ $$( 1 + T )^{2}$$
59$C_1$ $$( 1 + T )^{2}$$
good2$D_{4}$ $$1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4}$$
5$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
7$D_{4}$ $$1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
11$C_2^2$ $$1 + 17 T^{2} + p^{2} T^{4}$$
13$D_{4}$ $$1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 5 T + 13 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4}$$
31$C_4$ $$1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 15 T + 139 T^{2} + 15 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 13 T + 153 T^{2} + 13 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 8 T + 105 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 5 T + 141 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
79$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
83$D_{4}$ $$1 + T + 165 T^{2} + p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$