# Properties

 Label 2-177-59.58-c4-0-11 Degree $2$ Conductor $177$ Sign $0.586 + 0.810i$ 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 − 6.10i·2-s − 5.19·3-s − 21.2·4-s − 12.8·5-s + 31.7i·6-s + 4.61·7-s + 32.0i·8-s + 27·9-s + 78.4i·10-s + 159. i·11-s + 110.·12-s + 148. i·13-s − 28.1i·14-s + 66.7·15-s − 144.·16-s + 154.·17-s + ⋯
 L(s)  = 1 − 1.52i·2-s − 0.577·3-s − 1.32·4-s − 0.513·5-s + 0.880i·6-s + 0.0942·7-s + 0.500i·8-s + 0.333·9-s + 0.784i·10-s + 1.31i·11-s + 0.766·12-s + 0.881i·13-s − 0.143i·14-s + 0.296·15-s − 0.564·16-s + 0.534·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $0.586 + 0.810i$ 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: $$(2,\ 177,\ (\ :2),\ 0.586 + 0.810i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.139575311$$ $$L(\frac12)$$ $$\approx$$ $$1.139575311$$ $$L(3)$$ 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 + (-2.04e3 - 2.82e3i)T$$
good2 $$1 + 6.10iT - 16T^{2}$$
5 $$1 + 12.8T + 625T^{2}$$
7 $$1 - 4.61T + 2.40e3T^{2}$$
11 $$1 - 159. iT - 1.46e4T^{2}$$
13 $$1 - 148. iT - 2.85e4T^{2}$$
17 $$1 - 154.T + 8.35e4T^{2}$$
19 $$1 - 313.T + 1.30e5T^{2}$$
23 $$1 + 420. iT - 2.79e5T^{2}$$
29 $$1 - 1.42e3T + 7.07e5T^{2}$$
31 $$1 + 1.10e3iT - 9.23e5T^{2}$$
37 $$1 + 186. iT - 1.87e6T^{2}$$
41 $$1 - 435.T + 2.82e6T^{2}$$
43 $$1 - 2.36e3iT - 3.41e6T^{2}$$
47 $$1 - 2.36e3iT - 4.87e6T^{2}$$
53 $$1 - 3.94e3T + 7.89e6T^{2}$$
61 $$1 - 1.73e3iT - 1.38e7T^{2}$$
67 $$1 + 2.71e3iT - 2.01e7T^{2}$$
71 $$1 - 5.43e3T + 2.54e7T^{2}$$
73 $$1 + 2.07e3iT - 2.83e7T^{2}$$
79 $$1 - 1.07e4T + 3.89e7T^{2}$$
83 $$1 - 1.09e4iT - 4.74e7T^{2}$$
89 $$1 - 1.73e3iT - 6.27e7T^{2}$$
97 $$1 - 3.59e3iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$