# Properties

 Label 2-160-5.4-c3-0-17 Degree $2$ Conductor $160$ Sign $-0.998 + 0.0521i$ Analytic cond. $9.44030$ Root an. cond. $3.07250$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 8.57i·3-s + (−0.582 − 11.1i)5-s − 22.1i·7-s − 46.4·9-s + 27.1·11-s + 70.3i·13-s + (−95.7 + 4.99i)15-s − 73.3i·17-s + 110.·19-s − 189.·21-s + 107. i·23-s + (−124. + 13.0i)25-s + 167. i·27-s − 68.6·29-s − 137.·31-s + ⋯
 L(s)  = 1 − 1.64i·3-s + (−0.0521 − 0.998i)5-s − 1.19i·7-s − 1.72·9-s + 0.743·11-s + 1.50i·13-s + (−1.64 + 0.0859i)15-s − 1.04i·17-s + 1.32·19-s − 1.97·21-s + 0.977i·23-s + (−0.994 + 0.104i)25-s + 1.19i·27-s − 0.439·29-s − 0.794·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$160$$    =    $$2^{5} \cdot 5$$ Sign: $-0.998 + 0.0521i$ Analytic conductor: $$9.44030$$ Root analytic conductor: $$3.07250$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{160} (129, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 160,\ (\ :3/2),\ -0.998 + 0.0521i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.0388225 - 1.48909i$$ $$L(\frac12)$$ $$\approx$$ $$0.0388225 - 1.48909i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (0.582 + 11.1i)T$$
good3 $$1 + 8.57iT - 27T^{2}$$
7 $$1 + 22.1iT - 343T^{2}$$
11 $$1 - 27.1T + 1.33e3T^{2}$$
13 $$1 - 70.3iT - 2.19e3T^{2}$$
17 $$1 + 73.3iT - 4.91e3T^{2}$$
19 $$1 - 110.T + 6.85e3T^{2}$$
23 $$1 - 107. iT - 1.21e4T^{2}$$
29 $$1 + 68.6T + 2.43e4T^{2}$$
31 $$1 + 137.T + 2.97e4T^{2}$$
37 $$1 - 60.3iT - 5.06e4T^{2}$$
41 $$1 - 95.1T + 6.89e4T^{2}$$
43 $$1 + 501. iT - 7.95e4T^{2}$$
47 $$1 + 439. iT - 1.03e5T^{2}$$
53 $$1 - 286. iT - 1.48e5T^{2}$$
59 $$1 - 547.T + 2.05e5T^{2}$$
61 $$1 - 511.T + 2.26e5T^{2}$$
67 $$1 - 301. iT - 3.00e5T^{2}$$
71 $$1 - 82.8T + 3.57e5T^{2}$$
73 $$1 - 763. iT - 3.89e5T^{2}$$
79 $$1 + 1.01e3T + 4.93e5T^{2}$$
83 $$1 + 704. iT - 5.71e5T^{2}$$
89 $$1 - 743.T + 7.04e5T^{2}$$
97 $$1 + 1.13e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$