# Properties

 Label 2-177-177.176-c3-0-23 Degree $2$ Conductor $177$ Sign $0.213 + 0.976i$ Analytic cond. $10.4433$ Root an. cond. $3.23161$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−5.07 + 1.11i)3-s − 8·4-s + 22.0i·5-s − 34.4·7-s + (24.5 − 11.2i)9-s + (40.6 − 8.88i)12-s + (−24.4 − 111. i)15-s + 64·16-s + 61.4i·17-s + 40.2·19-s − 176. i·20-s + (174. − 38.2i)21-s − 360.·25-s + (−111. + 84.4i)27-s + 275.·28-s − 272. i·29-s + ⋯
 L(s)  = 1 + (−0.976 + 0.213i)3-s − 4-s + 1.97i·5-s − 1.86·7-s + (0.908 − 0.417i)9-s + (0.976 − 0.213i)12-s + (−0.421 − 1.92i)15-s + 16-s + 0.876i·17-s + 0.486·19-s − 1.97i·20-s + (1.81 − 0.397i)21-s − 2.88·25-s + (−0.798 + 0.602i)27-s + 1.86·28-s − 1.74i·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $0.213 + 0.976i$ Analytic conductor: $$10.4433$$ Root analytic conductor: $$3.23161$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{177} (176, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 177,\ (\ :3/2),\ 0.213 + 0.976i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.0207153 - 0.0166723i$$ $$L(\frac12)$$ $$\approx$$ $$0.0207153 - 0.0166723i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (5.07 - 1.11i)T$$
59 $$1 + 453. iT$$
good2 $$1 + 8T^{2}$$
5 $$1 - 22.0iT - 125T^{2}$$
7 $$1 + 34.4T + 343T^{2}$$
11 $$1 + 1.33e3T^{2}$$
13 $$1 - 2.19e3T^{2}$$
17 $$1 - 61.4iT - 4.91e3T^{2}$$
19 $$1 - 40.2T + 6.85e3T^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 + 272. iT - 2.43e4T^{2}$$
31 $$1 - 2.97e4T^{2}$$
37 $$1 - 5.06e4T^{2}$$
41 $$1 - 458. iT - 6.89e4T^{2}$$
43 $$1 - 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 - 632. iT - 1.48e5T^{2}$$
61 $$1 - 2.26e5T^{2}$$
67 $$1 - 3.00e5T^{2}$$
71 $$1 + 1.18e3iT - 3.57e5T^{2}$$
73 $$1 - 3.89e5T^{2}$$
79 $$1 + 560.T + 4.93e5T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + 7.04e5T^{2}$$
97 $$1 - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$