# Properties

 Label 2-177-3.2-c2-0-12 Degree $2$ Conductor $177$ Sign $-0.976 + 0.217i$ Analytic cond. $4.82290$ Root an. cond. $2.19611$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.79i·2-s + (0.651 + 2.92i)3-s − 3.81·4-s + 4.04i·5-s + (−8.18 + 1.82i)6-s + 6.95·7-s + 0.509i·8-s + (−8.15 + 3.81i)9-s − 11.3·10-s − 3.60i·11-s + (−2.48 − 11.1i)12-s + 6.03·13-s + 19.4i·14-s + (−11.8 + 2.63i)15-s − 16.6·16-s − 27.5i·17-s + ⋯
 L(s)  = 1 + 1.39i·2-s + (0.217 + 0.976i)3-s − 0.954·4-s + 0.808i·5-s + (−1.36 + 0.303i)6-s + 0.993·7-s + 0.0636i·8-s + (−0.905 + 0.423i)9-s − 1.13·10-s − 0.327i·11-s + (−0.207 − 0.931i)12-s + 0.463·13-s + 1.38i·14-s + (−0.789 + 0.175i)15-s − 1.04·16-s − 1.62i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $-0.976 + 0.217i$ 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: $$(2,\ 177,\ (\ :1),\ -0.976 + 0.217i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.182300 - 1.65998i$$ $$L(\frac12)$$ $$\approx$$ $$0.182300 - 1.65998i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.651 - 2.92i)T$$
59 $$1 + 7.68iT$$
good2 $$1 - 2.79iT - 4T^{2}$$
5 $$1 - 4.04iT - 25T^{2}$$
7 $$1 - 6.95T + 49T^{2}$$
11 $$1 + 3.60iT - 121T^{2}$$
13 $$1 - 6.03T + 169T^{2}$$
17 $$1 + 27.5iT - 289T^{2}$$
19 $$1 - 11.5T + 361T^{2}$$
23 $$1 - 6.63iT - 529T^{2}$$
29 $$1 + 7.59iT - 841T^{2}$$
31 $$1 - 40.0T + 961T^{2}$$
37 $$1 + 53.5T + 1.36e3T^{2}$$
41 $$1 - 33.0iT - 1.68e3T^{2}$$
43 $$1 + 19.3T + 1.84e3T^{2}$$
47 $$1 - 51.4iT - 2.20e3T^{2}$$
53 $$1 - 40.9iT - 2.80e3T^{2}$$
61 $$1 - 48.9T + 3.72e3T^{2}$$
67 $$1 - 85.3T + 4.48e3T^{2}$$
71 $$1 - 32.2iT - 5.04e3T^{2}$$
73 $$1 - 71.6T + 5.32e3T^{2}$$
79 $$1 + 85.8T + 6.24e3T^{2}$$
83 $$1 + 131. iT - 6.88e3T^{2}$$
89 $$1 - 96.1iT - 7.92e3T^{2}$$
97 $$1 + 80.8T + 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

−13.93412855019559765774762958467, −11.57803527058354110989532413886, −11.07061955784256383632760738088, −9.788511250940223609792103981599, −8.670335317977843355696857675980, −7.83827573707144839121365773146, −6.77152634788859122937962262723, −5.49521843333468464130186124480, −4.64425939036767659079000552477, −2.93094416412700754803819381035, 1.10922538316966126789794824765, 2.02792484230151286249533864160, 3.69883813627867991391803631952, 5.12952592119713461433428358457, 6.72488924293090972201610804296, 8.212858369023795067834565123958, 8.782047456079929974864771492753, 10.19007549063417340344684111765, 11.19593446572248560819502303321, 12.05122745732307749224625300764