# Properties

 Degree $2$ Conductor $177$ Sign $0.886 - 0.462i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.46·2-s + (1.16 + 1.28i)3-s + 0.139·4-s − 0.594i·5-s + (1.69 + 1.87i)6-s + 1.86·7-s − 2.72·8-s + (−0.300 + 2.98i)9-s − 0.869i·10-s − 0.676·11-s + (0.161 + 0.178i)12-s − 5.37i·13-s + 2.72·14-s + (0.763 − 0.690i)15-s − 4.25·16-s + 1.70i·17-s + ⋯
 L(s)  = 1 + 1.03·2-s + (0.670 + 0.741i)3-s + 0.0695·4-s − 0.265i·5-s + (0.693 + 0.767i)6-s + 0.703·7-s − 0.962·8-s + (−0.100 + 0.994i)9-s − 0.274i·10-s − 0.204·11-s + (0.0466 + 0.0516i)12-s − 1.49i·13-s + 0.727·14-s + (0.197 − 0.178i)15-s − 1.06·16-s + 0.412i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $0.886 - 0.462i$ Motivic weight: $$1$$ Character: $\chi_{177} (176, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 177,\ (\ :1/2),\ 0.886 - 0.462i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.98694 + 0.486685i$$ $$L(\frac12)$$ $$\approx$$ $$1.98694 + 0.486685i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.16 - 1.28i)T$$
59 $$1 + (1.93 - 7.43i)T$$
good2 $$1 - 1.46T + 2T^{2}$$
5 $$1 + 0.594iT - 5T^{2}$$
7 $$1 - 1.86T + 7T^{2}$$
11 $$1 + 0.676T + 11T^{2}$$
13 $$1 + 5.37iT - 13T^{2}$$
17 $$1 - 1.70iT - 17T^{2}$$
19 $$1 + 2.25T + 19T^{2}$$
23 $$1 + 4.86T + 23T^{2}$$
29 $$1 + 6.24iT - 29T^{2}$$
31 $$1 + 2.48iT - 31T^{2}$$
37 $$1 - 7.86iT - 37T^{2}$$
41 $$1 + 2.29iT - 41T^{2}$$
43 $$1 - 7.11iT - 43T^{2}$$
47 $$1 - 8.46T + 47T^{2}$$
53 $$1 + 5.73iT - 53T^{2}$$
61 $$1 - 10.0iT - 61T^{2}$$
67 $$1 - 5.37iT - 67T^{2}$$
71 $$1 - 5.92iT - 71T^{2}$$
73 $$1 + 8.26iT - 73T^{2}$$
79 $$1 - 8.29T + 79T^{2}$$
83 $$1 + 0.0941T + 83T^{2}$$
89 $$1 - 10.9T + 89T^{2}$$
97 $$1 - 17.8iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$