# Properties

 Label 2-19-1.1-c1-0-0 Degree $2$ Conductor $19$ Sign $1$ Analytic cond. $0.151715$ Root an. cond. $0.389507$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·3-s − 2·4-s + 3·5-s − 7-s + 9-s + 3·11-s + 4·12-s − 4·13-s − 6·15-s + 4·16-s − 3·17-s + 19-s − 6·20-s + 2·21-s + 4·25-s + 4·27-s + 2·28-s + 6·29-s − 4·31-s − 6·33-s − 3·35-s − 2·36-s + 2·37-s + 8·39-s − 6·41-s − 43-s − 6·44-s + ⋯
 L(s)  = 1 − 1.15·3-s − 4-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 1.10·13-s − 1.54·15-s + 16-s − 0.727·17-s + 0.229·19-s − 1.34·20-s + 0.436·21-s + 4/5·25-s + 0.769·27-s + 0.377·28-s + 1.11·29-s − 0.718·31-s − 1.04·33-s − 0.507·35-s − 1/3·36-s + 0.328·37-s + 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.904·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$19$$ Sign: $1$ Analytic conductor: $$0.151715$$ Root analytic conductor: $$0.389507$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 19,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4532532444$$ $$L(\frac12)$$ $$\approx$$ $$0.4532532444$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad19 $$1 - T$$
good2 $$1 + p T^{2}$$
3 $$1 + 2 T + p T^{2}$$
5 $$1 - 3 T + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 - 3 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + 3 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + T + p T^{2}$$
47 $$1 + 3 T + p T^{2}$$
53 $$1 - 12 T + p T^{2}$$
59 $$1 + 6 T + p T^{2}$$
61 $$1 + T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 6 T + p T^{2}$$
73 $$1 + 7 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 - 12 T + p T^{2}$$
97 $$1 - 8 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$