# Properties

 Degree $2$ Conductor $33$ Sign $0.503 + 0.863i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.57i·2-s − 5.19·3-s + 13.5·4-s + 15.5·5-s + 8.18i·6-s − 93.8i·7-s − 46.4i·8-s + 27·9-s − 24.4i·10-s + (60.9 + 104. i)11-s − 70.2·12-s + 29.4i·13-s − 147.·14-s − 80.8·15-s + 143.·16-s + 251. i·17-s + ⋯
 L(s)  = 1 − 0.393i·2-s − 0.577·3-s + 0.845·4-s + 0.622·5-s + 0.227i·6-s − 1.91i·7-s − 0.726i·8-s + 0.333·9-s − 0.244i·10-s + (0.503 + 0.863i)11-s − 0.487·12-s + 0.174i·13-s − 0.753·14-s − 0.359·15-s + 0.559·16-s + 0.871i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$33$$    =    $$3 \cdot 11$$ Sign: $0.503 + 0.863i$ Motivic weight: $$4$$ Character: $\chi_{33} (10, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 33,\ (\ :2),\ 0.503 + 0.863i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.33134 - 0.764949i$$ $$L(\frac12)$$ $$\approx$$ $$1.33134 - 0.764949i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 5.19T$$
11 $$1 + (-60.9 - 104. i)T$$
good2 $$1 + 1.57iT - 16T^{2}$$
5 $$1 - 15.5T + 625T^{2}$$
7 $$1 + 93.8iT - 2.40e3T^{2}$$
13 $$1 - 29.4iT - 2.85e4T^{2}$$
17 $$1 - 251. iT - 8.35e4T^{2}$$
19 $$1 + 80.2iT - 1.30e5T^{2}$$
23 $$1 + 702.T + 2.79e5T^{2}$$
29 $$1 - 1.44e3iT - 7.07e5T^{2}$$
31 $$1 - 1.27e3T + 9.23e5T^{2}$$
37 $$1 - 115.T + 1.87e6T^{2}$$
41 $$1 + 1.07e3iT - 2.82e6T^{2}$$
43 $$1 - 1.88e3iT - 3.41e6T^{2}$$
47 $$1 - 1.59e3T + 4.87e6T^{2}$$
53 $$1 - 1.19e3T + 7.89e6T^{2}$$
59 $$1 - 1.89e3T + 1.21e7T^{2}$$
61 $$1 - 3.77e3iT - 1.38e7T^{2}$$
67 $$1 + 1.25e3T + 2.01e7T^{2}$$
71 $$1 + 3.59e3T + 2.54e7T^{2}$$
73 $$1 - 1.75e3iT - 2.83e7T^{2}$$
79 $$1 + 6.74e3iT - 3.89e7T^{2}$$
83 $$1 + 8.61e3iT - 4.74e7T^{2}$$
89 $$1 + 9.33e3T + 6.27e7T^{2}$$
97 $$1 - 1.10e4T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$