# Properties

 Label 2-33-33.32-c5-0-9 Degree $2$ Conductor $33$ Sign $0.622 - 0.782i$ 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 + 10.1·2-s + (−10.9 + 11.0i)3-s + 70.2·4-s + 88.5i·5-s + (−110. + 112. i)6-s − 126. i·7-s + 387.·8-s + (−2.70 − 242. i)9-s + 895. i·10-s + (398. − 43.3i)11-s + (−770. + 779. i)12-s − 523. i·13-s − 1.27e3i·14-s + (−981. − 970. i)15-s + 1.66e3·16-s − 1.03e3·17-s + ⋯
 L(s)  = 1 + 1.78·2-s + (−0.703 + 0.711i)3-s + 2.19·4-s + 1.58i·5-s + (−1.25 + 1.27i)6-s − 0.973i·7-s + 2.13·8-s + (−0.0111 − 0.999i)9-s + 2.83i·10-s + (0.994 − 0.108i)11-s + (−1.54 + 1.56i)12-s − 0.858i·13-s − 1.74i·14-s + (−1.12 − 1.11i)15-s + 1.62·16-s − 0.872·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$33$$    =    $$3 \cdot 11$$ Sign: $0.622 - 0.782i$ 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: $$(2,\ 33,\ (\ :5/2),\ 0.622 - 0.782i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.99626 + 1.44609i$$ $$L(\frac12)$$ $$\approx$$ $$2.99626 + 1.44609i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (10.9 - 11.0i)T$$
11 $$1 + (-398. + 43.3i)T$$
good2 $$1 - 10.1T + 32T^{2}$$
5 $$1 - 88.5iT - 3.12e3T^{2}$$
7 $$1 + 126. iT - 1.68e4T^{2}$$
13 $$1 + 523. iT - 3.71e5T^{2}$$
17 $$1 + 1.03e3T + 1.41e6T^{2}$$
19 $$1 - 842. iT - 2.47e6T^{2}$$
23 $$1 + 2.47e3iT - 6.43e6T^{2}$$
29 $$1 - 3.56e3T + 2.05e7T^{2}$$
31 $$1 + 215.T + 2.86e7T^{2}$$
37 $$1 + 3.62e3T + 6.93e7T^{2}$$
41 $$1 + 8.84e3T + 1.15e8T^{2}$$
43 $$1 - 5.26e3iT - 1.47e8T^{2}$$
47 $$1 - 9.05e3iT - 2.29e8T^{2}$$
53 $$1 - 897. iT - 4.18e8T^{2}$$
59 $$1 - 3.62e4iT - 7.14e8T^{2}$$
61 $$1 + 1.53e4iT - 8.44e8T^{2}$$
67 $$1 + 5.33e4T + 1.35e9T^{2}$$
71 $$1 + 3.36e4iT - 1.80e9T^{2}$$
73 $$1 + 7.61e4iT - 2.07e9T^{2}$$
79 $$1 - 1.06e5iT - 3.07e9T^{2}$$
83 $$1 - 2.03e4T + 3.93e9T^{2}$$
89 $$1 + 1.18e4iT - 5.58e9T^{2}$$
97 $$1 + 6.77e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$