# Properties

 Label 4-539e2-1.1-c3e2-0-0 Degree $4$ Conductor $290521$ Sign $1$ Analytic cond. $1011.36$ Root an. cond. $5.63932$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·3-s − 10·4-s − 2·5-s + 4·6-s − 32·8-s − 3·9-s − 4·10-s − 22·11-s − 20·12-s − 80·13-s − 4·15-s + 44·16-s + 124·17-s − 6·18-s − 72·19-s + 20·20-s − 44·22-s − 98·23-s − 64·24-s − 55·25-s − 160·26-s + 34·27-s + 144·29-s − 8·30-s + 34·31-s + 248·32-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.384·3-s − 5/4·4-s − 0.178·5-s + 0.272·6-s − 1.41·8-s − 1/9·9-s − 0.126·10-s − 0.603·11-s − 0.481·12-s − 1.70·13-s − 0.0688·15-s + 0.687·16-s + 1.76·17-s − 0.0785·18-s − 0.869·19-s + 0.223·20-s − 0.426·22-s − 0.888·23-s − 0.544·24-s − 0.439·25-s − 1.20·26-s + 0.242·27-s + 0.922·29-s − 0.0486·30-s + 0.196·31-s + 1.37·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$290521$$    =    $$7^{4} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$1011.36$$ Root analytic conductor: $$5.63932$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{539} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 290521,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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)$
bad7 $$1$$
11$C_1$ $$( 1 + p T )^{2}$$
good2$D_{4}$ $$1 - p T + 7 p T^{2} - p^{4} T^{3} + p^{6} T^{4}$$
3$D_{4}$ $$1 - 2 T + 7 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4}$$
5$D_{4}$ $$1 + 2 T + 59 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 - 124 T + 13238 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 72 T + 4214 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 98 T + 22847 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 144 T + 44554 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 34 T + 57519 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 54 T + 101843 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 536 T + 209618 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 + 60 T + 159146 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 - 272 T + 182942 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 492 T + 348862 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 + 634 T + 458975 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 840 T + 528794 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 754 T + 742455 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 678 T + 813415 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 400 T + 160962 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 4 p T - 279966 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 468 T + 1155130 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 1842 T + 1935427 T^{2} - 1842 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 2194 T + 2966547 T^{2} + 2194 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$