# Properties

 Degree $2$ Conductor $89$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 4-s − 5-s + 6-s − 4·7-s + 3·8-s − 2·9-s + 10-s − 2·11-s + 12-s + 2·13-s + 4·14-s + 15-s − 16-s + 3·17-s + 2·18-s − 5·19-s + 20-s + 4·21-s + 2·22-s + 7·23-s − 3·24-s − 4·25-s − 2·26-s + 5·27-s + 4·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.14·19-s + 0.223·20-s + 0.872·21-s + 0.426·22-s + 1.45·23-s − 0.612·24-s − 4/5·25-s − 0.392·26-s + 0.962·27-s + 0.755·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$89$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{89} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 89,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad89 $$1 + T$$
good2 $$1 + T + p T^{2}$$
3 $$1 + T + p T^{2}$$
5 $$1 + T + p T^{2}$$
7 $$1 + 4 T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 3 T + p T^{2}$$
19 $$1 + 5 T + p T^{2}$$
23 $$1 - 7 T + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + 9 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 7 T + p T^{2}$$
47 $$1 + 12 T + p T^{2}$$
53 $$1 + 3 T + p T^{2}$$
59 $$1 - 4 T + p T^{2}$$
61 $$1 - 6 T + p T^{2}$$
67 $$1 - 12 T + p T^{2}$$
71 $$1 + 10 T + p T^{2}$$
73 $$1 - 7 T + p T^{2}$$
79 $$1 + 6 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
97 $$1 - 9 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$