# Properties

 Label 2-33-33.32-c5-0-8 Degree $2$ Conductor $33$ Sign $0.200 + 0.979i$ Analytic cond. $5.29266$ Root an. cond. $2.30057$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 8.47·2-s + (11.5 − 10.4i)3-s + 39.8·4-s + 35.7i·5-s + (−98.3 + 88.3i)6-s − 11.5i·7-s − 66.9·8-s + (25.9 − 241. i)9-s − 302. i·10-s + (202. − 346. i)11-s + (462. − 415. i)12-s − 933. i·13-s + 98.0i·14-s + (371. + 414. i)15-s − 708.·16-s + 25.0·17-s + ⋯
 L(s)  = 1 − 1.49·2-s + (0.743 − 0.668i)3-s + 1.24·4-s + 0.638i·5-s + (−1.11 + 1.00i)6-s − 0.0892i·7-s − 0.369·8-s + (0.106 − 0.994i)9-s − 0.957i·10-s + (0.505 − 0.862i)11-s + (0.927 − 0.833i)12-s − 1.53i·13-s + 0.133i·14-s + (0.426 + 0.475i)15-s − 0.692·16-s + 0.0210·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$33$$    =    $$3 \cdot 11$$ Sign: $0.200 + 0.979i$ Analytic conductor: $$5.29266$$ Root analytic conductor: $$2.30057$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{33} (32, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 33,\ (\ :5/2),\ 0.200 + 0.979i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.688370 - 0.561667i$$ $$L(\frac12)$$ $$\approx$$ $$0.688370 - 0.561667i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-11.5 + 10.4i)T$$
11 $$1 + (-202. + 346. i)T$$
good2 $$1 + 8.47T + 32T^{2}$$
5 $$1 - 35.7iT - 3.12e3T^{2}$$
7 $$1 + 11.5iT - 1.68e4T^{2}$$
13 $$1 + 933. iT - 3.71e5T^{2}$$
17 $$1 - 25.0T + 1.41e6T^{2}$$
19 $$1 + 743. iT - 2.47e6T^{2}$$
23 $$1 + 594. iT - 6.43e6T^{2}$$
29 $$1 - 3.97e3T + 2.05e7T^{2}$$
31 $$1 + 3.07e3T + 2.86e7T^{2}$$
37 $$1 - 1.03e4T + 6.93e7T^{2}$$
41 $$1 + 1.77e4T + 1.15e8T^{2}$$
43 $$1 - 2.29e4iT - 1.47e8T^{2}$$
47 $$1 + 3.40e3iT - 2.29e8T^{2}$$
53 $$1 - 1.23e4iT - 4.18e8T^{2}$$
59 $$1 + 3.03e4iT - 7.14e8T^{2}$$
61 $$1 + 3.64e4iT - 8.44e8T^{2}$$
67 $$1 + 2.85e4T + 1.35e9T^{2}$$
71 $$1 - 5.93e4iT - 1.80e9T^{2}$$
73 $$1 - 7.24e4iT - 2.07e9T^{2}$$
79 $$1 + 2.69e4iT - 3.07e9T^{2}$$
83 $$1 + 3.24e4T + 3.93e9T^{2}$$
89 $$1 - 1.08e4iT - 5.58e9T^{2}$$
97 $$1 + 1.05e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$