# Properties

 Label 4-73e2-1.1-c1e2-0-0 Degree $4$ Conductor $5329$ Sign $1$ Analytic cond. $0.339781$ Root an. cond. $0.763484$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 3-s − 5-s + 6-s − 2·7-s + 8-s − 2·9-s − 10-s + 7·11-s − 13-s − 2·14-s − 15-s − 16-s − 4·17-s − 2·18-s − 14·19-s − 2·21-s + 7·22-s + 13·23-s + 24-s − 6·25-s − 26-s − 2·27-s + 2·29-s − 30-s + 6·31-s − 6·32-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 2.11·11-s − 0.277·13-s − 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.970·17-s − 0.471·18-s − 3.21·19-s − 0.436·21-s + 1.49·22-s + 2.71·23-s + 0.204·24-s − 6/5·25-s − 0.196·26-s − 0.384·27-s + 0.371·29-s − 0.182·30-s + 1.07·31-s − 1.06·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$5329$$    =    $$73^{2}$$ Sign: $1$ Analytic conductor: $$0.339781$$ Root analytic conductor: $$0.763484$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 5329,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.187013242$$ $$L(\frac12)$$ $$\approx$$ $$1.187013242$$ $$L(\frac{3}{2})$$ 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)$
bad73$C_1$ $$( 1 - T )^{2}$$
good2$D_{4}$ $$1 - T + T^{2} - p T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 - T + p T^{2} - p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4}$$
7$C_2$ $$( 1 + T + p T^{2} )^{2}$$
11$D_{4}$ $$1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
19$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
23$D_{4}$ $$1 - 13 T + 85 T^{2} - 13 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 8 T + 77 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
43$D_{4}$ $$1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4}$$
47$C_2$ $$( 1 - 9 T + p T^{2} )^{2}$$
53$D_{4}$ $$1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$D_{4}$ $$1 + 9 T + 139 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 3 T + 115 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - T + 129 T^{2} - p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 7 T + 97 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 5 T + 171 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$