# Properties

 Label 2-177-59.58-c4-0-13 Degree $2$ Conductor $177$ Sign $-0.985 + 0.169i$ 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 + 6.76i·2-s + 5.19·3-s − 29.7·4-s + 6.77·5-s + 35.1i·6-s + 45.8·7-s − 93.3i·8-s + 27·9-s + 45.8i·10-s + 152. i·11-s − 154.·12-s + 138. i·13-s + 310. i·14-s + 35.2·15-s + 155.·16-s + 61.1·17-s + ⋯
 L(s)  = 1 + 1.69i·2-s + 0.577·3-s − 1.86·4-s + 0.271·5-s + 0.976i·6-s + 0.936·7-s − 1.45i·8-s + 0.333·9-s + 0.458i·10-s + 1.25i·11-s − 1.07·12-s + 0.819i·13-s + 1.58i·14-s + 0.156·15-s + 0.605·16-s + 0.211·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $-0.985 + 0.169i$ 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: $$(2,\ 177,\ (\ :2),\ -0.985 + 0.169i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$2.113870443$$ $$L(\frac12)$$ $$\approx$$ $$2.113870443$$ $$L(3)$$ 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 + (-3.43e3 + 589. i)T$$
good2 $$1 - 6.76iT - 16T^{2}$$
5 $$1 - 6.77T + 625T^{2}$$
7 $$1 - 45.8T + 2.40e3T^{2}$$
11 $$1 - 152. iT - 1.46e4T^{2}$$
13 $$1 - 138. iT - 2.85e4T^{2}$$
17 $$1 - 61.1T + 8.35e4T^{2}$$
19 $$1 + 184.T + 1.30e5T^{2}$$
23 $$1 - 55.4iT - 2.79e5T^{2}$$
29 $$1 - 290.T + 7.07e5T^{2}$$
31 $$1 - 681. iT - 9.23e5T^{2}$$
37 $$1 - 1.42e3iT - 1.87e6T^{2}$$
41 $$1 - 1.53e3T + 2.82e6T^{2}$$
43 $$1 + 404. iT - 3.41e6T^{2}$$
47 $$1 + 1.60e3iT - 4.87e6T^{2}$$
53 $$1 + 1.32e3T + 7.89e6T^{2}$$
61 $$1 + 2.73e3iT - 1.38e7T^{2}$$
67 $$1 - 2.22e3iT - 2.01e7T^{2}$$
71 $$1 + 6.15e3T + 2.54e7T^{2}$$
73 $$1 - 2.81e3iT - 2.83e7T^{2}$$
79 $$1 - 7.74e3T + 3.89e7T^{2}$$
83 $$1 + 648. iT - 4.74e7T^{2}$$
89 $$1 + 5.75e3iT - 6.27e7T^{2}$$
97 $$1 + 1.51e3iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$