# Properties

 Label 2-177-59.58-c8-0-36 Degree $2$ Conductor $177$ Sign $-0.303 + 0.952i$ Analytic cond. $72.1060$ Root an. cond. $8.49152$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 25.6i·2-s + 46.7·3-s − 400.·4-s − 239.·5-s − 1.19e3i·6-s − 1.75e3·7-s + 3.71e3i·8-s + 2.18e3·9-s + 6.14e3i·10-s + 3.30e3i·11-s − 1.87e4·12-s + 1.31e4i·13-s + 4.49e4i·14-s − 1.12e4·15-s − 7.47e3·16-s + 1.19e5·17-s + ⋯
 L(s)  = 1 − 1.60i·2-s + 0.577·3-s − 1.56·4-s − 0.383·5-s − 0.924i·6-s − 0.731·7-s + 0.906i·8-s + 0.333·9-s + 0.614i·10-s + 0.225i·11-s − 0.904·12-s + 0.460i·13-s + 1.17i·14-s − 0.221·15-s − 0.114·16-s + 1.43·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $-0.303 + 0.952i$ Analytic conductor: $$72.1060$$ Root analytic conductor: $$8.49152$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{177} (58, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 177,\ (\ :4),\ -0.303 + 0.952i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$2.044249476$$ $$L(\frac12)$$ $$\approx$$ $$2.044249476$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 46.7T$$
59 $$1 + (-3.67e6 + 1.15e7i)T$$
good2 $$1 + 25.6iT - 256T^{2}$$
5 $$1 + 239.T + 3.90e5T^{2}$$
7 $$1 + 1.75e3T + 5.76e6T^{2}$$
11 $$1 - 3.30e3iT - 2.14e8T^{2}$$
13 $$1 - 1.31e4iT - 8.15e8T^{2}$$
17 $$1 - 1.19e5T + 6.97e9T^{2}$$
19 $$1 - 1.48e5T + 1.69e10T^{2}$$
23 $$1 - 1.44e5iT - 7.83e10T^{2}$$
29 $$1 - 3.97e5T + 5.00e11T^{2}$$
31 $$1 - 1.82e6iT - 8.52e11T^{2}$$
37 $$1 + 2.78e5iT - 3.51e12T^{2}$$
41 $$1 - 1.92e6T + 7.98e12T^{2}$$
43 $$1 + 3.71e6iT - 1.16e13T^{2}$$
47 $$1 + 4.04e6iT - 2.38e13T^{2}$$
53 $$1 + 1.28e6T + 6.22e13T^{2}$$
61 $$1 + 2.00e7iT - 1.91e14T^{2}$$
67 $$1 + 6.13e6iT - 4.06e14T^{2}$$
71 $$1 - 3.64e7T + 6.45e14T^{2}$$
73 $$1 + 1.88e7iT - 8.06e14T^{2}$$
79 $$1 - 7.26e7T + 1.51e15T^{2}$$
83 $$1 - 5.12e7iT - 2.25e15T^{2}$$
89 $$1 - 4.51e7iT - 3.93e15T^{2}$$
97 $$1 - 6.88e7iT - 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$