# Properties

 Label 2-32487-1.1-c1-0-3 Degree $2$ Conductor $32487$ Sign $1$ Analytic cond. $259.410$ Root an. cond. $16.1062$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2·2-s − 3-s + 2·4-s − 2·5-s − 2·6-s + 9-s − 4·10-s + 5·11-s − 2·12-s − 13-s + 2·15-s − 4·16-s − 17-s + 2·18-s − 6·19-s − 4·20-s + 10·22-s − 4·23-s − 25-s − 2·26-s − 27-s + 8·29-s + 4·30-s + 31-s − 8·32-s − 5·33-s − 2·34-s + ⋯
 L(s)  = 1 + 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s − 0.816·6-s + 1/3·9-s − 1.26·10-s + 1.50·11-s − 0.577·12-s − 0.277·13-s + 0.516·15-s − 16-s − 0.242·17-s + 0.471·18-s − 1.37·19-s − 0.894·20-s + 2.13·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.48·29-s + 0.730·30-s + 0.179·31-s − 1.41·32-s − 0.870·33-s − 0.342·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$32487$$    =    $$3 \cdot 7^{2} \cdot 13 \cdot 17$$ Sign: $1$ Analytic conductor: $$259.410$$ Root analytic conductor: $$16.1062$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 32487,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.276620091$$ $$L(\frac12)$$ $$\approx$$ $$2.276620091$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

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