# Properties

 Label 2-165-5.4-c5-0-0 Degree $2$ Conductor $165$ Sign $0.398 + 0.917i$ Analytic cond. $26.4633$ Root an. cond. $5.14425$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 8.57i·2-s − 9i·3-s − 41.5·4-s + (−51.2 + 22.2i)5-s + 77.1·6-s + 178. i·7-s − 81.7i·8-s − 81·9-s + (−191. − 439. i)10-s + 121·11-s + 373. i·12-s + 361. i·13-s − 1.53e3·14-s + (200. + 461. i)15-s − 627.·16-s + 934. i·17-s + ⋯
 L(s)  = 1 + 1.51i·2-s − 0.577i·3-s − 1.29·4-s + (−0.917 + 0.398i)5-s + 0.875·6-s + 1.37i·7-s − 0.451i·8-s − 0.333·9-s + (−0.604 − 1.39i)10-s + 0.301·11-s + 0.749i·12-s + 0.594i·13-s − 2.08·14-s + (0.230 + 0.529i)15-s − 0.613·16-s + 0.783i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$165$$    =    $$3 \cdot 5 \cdot 11$$ Sign: $0.398 + 0.917i$ Analytic conductor: $$26.4633$$ Root analytic conductor: $$5.14425$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{165} (34, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 165,\ (\ :5/2),\ 0.398 + 0.917i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.3200166294$$ $$L(\frac12)$$ $$\approx$$ $$0.3200166294$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 9iT$$
5 $$1 + (51.2 - 22.2i)T$$
11 $$1 - 121T$$
good2 $$1 - 8.57iT - 32T^{2}$$
7 $$1 - 178. iT - 1.68e4T^{2}$$
13 $$1 - 361. iT - 3.71e5T^{2}$$
17 $$1 - 934. iT - 1.41e6T^{2}$$
19 $$1 + 753.T + 2.47e6T^{2}$$
23 $$1 + 3.23e3iT - 6.43e6T^{2}$$
29 $$1 + 2.60e3T + 2.05e7T^{2}$$
31 $$1 - 662.T + 2.86e7T^{2}$$
37 $$1 + 1.29e4iT - 6.93e7T^{2}$$
41 $$1 + 2.53e3T + 1.15e8T^{2}$$
43 $$1 + 2.20e4iT - 1.47e8T^{2}$$
47 $$1 - 2.08e4iT - 2.29e8T^{2}$$
53 $$1 - 2.74e4iT - 4.18e8T^{2}$$
59 $$1 + 7.75e3T + 7.14e8T^{2}$$
61 $$1 - 3.84e4T + 8.44e8T^{2}$$
67 $$1 - 3.55e4iT - 1.35e9T^{2}$$
71 $$1 + 6.26e4T + 1.80e9T^{2}$$
73 $$1 + 6.89e4iT - 2.07e9T^{2}$$
79 $$1 - 1.73e4T + 3.07e9T^{2}$$
83 $$1 - 8.90e4iT - 3.93e9T^{2}$$
89 $$1 + 1.29e5T + 5.58e9T^{2}$$
97 $$1 + 1.36e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$