# Properties

 Label 2-177-59.58-c4-0-32 Degree $2$ Conductor $177$ Sign $-0.989 - 0.142i$ Analytic cond. $18.2964$ Root an. cond. $4.27743$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4.05i·2-s − 5.19·3-s − 0.408·4-s − 16.4·5-s + 21.0i·6-s + 47.3·7-s − 63.1i·8-s + 27·9-s + 66.4i·10-s − 25.6i·11-s + 2.12·12-s − 105. i·13-s − 191. i·14-s + 85.2·15-s − 262.·16-s + 441.·17-s + ⋯
 L(s)  = 1 − 1.01i·2-s − 0.577·3-s − 0.0255·4-s − 0.656·5-s + 0.584i·6-s + 0.965·7-s − 0.986i·8-s + 0.333·9-s + 0.664i·10-s − 0.211i·11-s + 0.0147·12-s − 0.624i·13-s − 0.977i·14-s + 0.378·15-s − 1.02·16-s + 1.52·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $-0.989 - 0.142i$ Analytic conductor: $$18.2964$$ Root analytic conductor: $$4.27743$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{177} (58, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 177,\ (\ :2),\ -0.989 - 0.142i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.093068079$$ $$L(\frac12)$$ $$\approx$$ $$1.093068079$$ $$L(3)$$ 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$$
59 $$1 + (3.44e3 + 496. i)T$$
good2 $$1 + 4.05iT - 16T^{2}$$
5 $$1 + 16.4T + 625T^{2}$$
7 $$1 - 47.3T + 2.40e3T^{2}$$
11 $$1 + 25.6iT - 1.46e4T^{2}$$
13 $$1 + 105. iT - 2.85e4T^{2}$$
17 $$1 - 441.T + 8.35e4T^{2}$$
19 $$1 + 560.T + 1.30e5T^{2}$$
23 $$1 + 764. iT - 2.79e5T^{2}$$
29 $$1 + 1.38e3T + 7.07e5T^{2}$$
31 $$1 - 950. iT - 9.23e5T^{2}$$
37 $$1 + 632. iT - 1.87e6T^{2}$$
41 $$1 - 1.02e3T + 2.82e6T^{2}$$
43 $$1 - 2.95e3iT - 3.41e6T^{2}$$
47 $$1 + 4.32e3iT - 4.87e6T^{2}$$
53 $$1 + 2.11e3T + 7.89e6T^{2}$$
61 $$1 + 4.27e3iT - 1.38e7T^{2}$$
67 $$1 - 867. iT - 2.01e7T^{2}$$
71 $$1 + 236.T + 2.54e7T^{2}$$
73 $$1 - 6.21e3iT - 2.83e7T^{2}$$
79 $$1 - 8.91e3T + 3.89e7T^{2}$$
83 $$1 + 1.08e4iT - 4.74e7T^{2}$$
89 $$1 + 6.26e3iT - 6.27e7T^{2}$$
97 $$1 + 7.89e3iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$