# Properties

 Label 2-177-177.176-c5-0-27 Degree $2$ Conductor $177$ Sign $0.983 + 0.183i$ Analytic cond. $28.3879$ Root an. cond. $5.32803$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.86 − 15.3i)3-s − 32·4-s + 49.9i·5-s − 258.·7-s + (−226. − 87.6i)9-s + (−91.5 + 490. i)12-s + (764. + 142. i)15-s + 1.02e3·16-s − 683. i·17-s + 896.·19-s − 1.59e3i·20-s + (−739. + 3.96e3i)21-s + 633.·25-s + (−1.99e3 + 3.22e3i)27-s + 8.27e3·28-s + 3.24e3i·29-s + ⋯
 L(s)  = 1 + (0.183 − 0.983i)3-s − 4-s + 0.892i·5-s − 1.99·7-s + (−0.932 − 0.360i)9-s + (−0.183 + 0.983i)12-s + (0.877 + 0.163i)15-s + 16-s − 0.573i·17-s + 0.569·19-s − 0.892i·20-s + (−0.366 + 1.96i)21-s + 0.202·25-s + (−0.525 + 0.850i)27-s + 1.99·28-s + 0.716i·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $0.983 + 0.183i$ Analytic conductor: $$28.3879$$ Root analytic conductor: $$5.32803$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{177} (176, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 177,\ (\ :5/2),\ 0.983 + 0.183i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.8339405406$$ $$L(\frac12)$$ $$\approx$$ $$0.8339405406$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-2.86 + 15.3i)T$$
59 $$1 + 2.67e4iT$$
good2 $$1 + 32T^{2}$$
5 $$1 - 49.9iT - 3.12e3T^{2}$$
7 $$1 + 258.T + 1.68e4T^{2}$$
11 $$1 + 1.61e5T^{2}$$
13 $$1 - 3.71e5T^{2}$$
17 $$1 + 683. iT - 1.41e6T^{2}$$
19 $$1 - 896.T + 2.47e6T^{2}$$
23 $$1 + 6.43e6T^{2}$$
29 $$1 - 3.24e3iT - 2.05e7T^{2}$$
31 $$1 - 2.86e7T^{2}$$
37 $$1 - 6.93e7T^{2}$$
41 $$1 + 1.89e4iT - 1.15e8T^{2}$$
43 $$1 - 1.47e8T^{2}$$
47 $$1 + 2.29e8T^{2}$$
53 $$1 - 3.88e4iT - 4.18e8T^{2}$$
61 $$1 - 8.44e8T^{2}$$
67 $$1 - 1.35e9T^{2}$$
71 $$1 - 5.94e4iT - 1.80e9T^{2}$$
73 $$1 - 2.07e9T^{2}$$
79 $$1 - 9.68e4T + 3.07e9T^{2}$$
83 $$1 + 3.93e9T^{2}$$
89 $$1 + 5.58e9T^{2}$$
97 $$1 - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$