# Properties

 Label 2-177-3.2-c2-0-3 Degree $2$ Conductor $177$ Sign $-0.987 + 0.158i$ Analytic cond. $4.82290$ Root an. cond. $2.19611$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2.97i·2-s + (−0.474 − 2.96i)3-s − 4.85·4-s + 3.00i·5-s + (8.81 − 1.41i)6-s − 3.67·7-s − 2.55i·8-s + (−8.54 + 2.81i)9-s − 8.94·10-s + 14.3i·11-s + (2.30 + 14.3i)12-s − 8.01·13-s − 10.9i·14-s + (8.90 − 1.42i)15-s − 11.8·16-s + 0.828i·17-s + ⋯
 L(s)  = 1 + 1.48i·2-s + (−0.158 − 0.987i)3-s − 1.21·4-s + 0.601i·5-s + (1.46 − 0.235i)6-s − 0.525·7-s − 0.319i·8-s + (−0.949 + 0.312i)9-s − 0.894·10-s + 1.30i·11-s + (0.192 + 1.19i)12-s − 0.616·13-s − 0.782i·14-s + (0.593 − 0.0951i)15-s − 0.739·16-s + 0.0487i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $-0.987 + 0.158i$ Analytic conductor: $$4.82290$$ Root analytic conductor: $$2.19611$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{177} (119, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 177,\ (\ :1),\ -0.987 + 0.158i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.0605743 - 0.760688i$$ $$L(\frac12)$$ $$\approx$$ $$0.0605743 - 0.760688i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (0.474 + 2.96i)T$$
59 $$1 - 7.68iT$$
good2 $$1 - 2.97iT - 4T^{2}$$
5 $$1 - 3.00iT - 25T^{2}$$
7 $$1 + 3.67T + 49T^{2}$$
11 $$1 - 14.3iT - 121T^{2}$$
13 $$1 + 8.01T + 169T^{2}$$
17 $$1 - 0.828iT - 289T^{2}$$
19 $$1 + 25.2T + 361T^{2}$$
23 $$1 - 7.83iT - 529T^{2}$$
29 $$1 + 2.34iT - 841T^{2}$$
31 $$1 - 25.4T + 961T^{2}$$
37 $$1 + 0.422T + 1.36e3T^{2}$$
41 $$1 - 6.28iT - 1.68e3T^{2}$$
43 $$1 - 48.4T + 1.84e3T^{2}$$
47 $$1 - 63.1iT - 2.20e3T^{2}$$
53 $$1 + 24.6iT - 2.80e3T^{2}$$
61 $$1 - 73.6T + 3.72e3T^{2}$$
67 $$1 + 100.T + 4.48e3T^{2}$$
71 $$1 - 47.9iT - 5.04e3T^{2}$$
73 $$1 - 130.T + 5.32e3T^{2}$$
79 $$1 + 76.0T + 6.24e3T^{2}$$
83 $$1 - 142. iT - 6.88e3T^{2}$$
89 $$1 + 73.4iT - 7.92e3T^{2}$$
97 $$1 - 84.6T + 9.40e3T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.06458642522607676791569507670, −12.36476677066018199045201116201, −11.05663142652024460867142561689, −9.727328620440980442938539606704, −8.462478430323792589534220894018, −7.42422597692704693681265563257, −6.82265665342073600474998456198, −6.05134840998559125417098685960, −4.68402875119699069604619588504, −2.43139976637656382833909849441, 0.44607198403415218685295639086, 2.69732905977761190226735165472, 3.82701923130854466453054817910, 4.90310215219201926220468544187, 6.32428249889824380664428925610, 8.522946106869642847472270537503, 9.188924226912385438965094072230, 10.23083618223670774313342914269, 10.85021776480335463447202641572, 11.80297106107553639254317414455