# Properties

 Label 2-177-59.58-c10-0-13 Degree $2$ Conductor $177$ Sign $0.515 + 0.857i$ Analytic cond. $112.458$ Root an. cond. $10.6046$ Motivic weight $10$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 48.0i·2-s − 140.·3-s − 1.28e3·4-s − 1.21e3·5-s + 6.74e3i·6-s − 4.24e3·7-s + 1.26e4i·8-s + 1.96e4·9-s + 5.82e4i·10-s − 3.39e4i·11-s + 1.80e5·12-s − 4.13e5i·13-s + 2.03e5i·14-s + 1.70e5·15-s − 7.10e5·16-s − 2.03e6·17-s + ⋯
 L(s)  = 1 − 1.50i·2-s − 0.577·3-s − 1.25·4-s − 0.388·5-s + 0.867i·6-s − 0.252·7-s + 0.384i·8-s + 0.333·9-s + 0.582i·10-s − 0.210i·11-s + 0.725·12-s − 1.11i·13-s + 0.378i·14-s + 0.224·15-s − 0.678·16-s − 1.43·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $0.515 + 0.857i$ Analytic conductor: $$112.458$$ Root analytic conductor: $$10.6046$$ Motivic weight: $$10$$ Rational: no Arithmetic: yes Character: $\chi_{177} (58, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 177,\ (\ :5),\ 0.515 + 0.857i)$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$0.4977827760$$ $$L(\frac12)$$ $$\approx$$ $$0.4977827760$$ $$L(6)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 140.T$$
59 $$1 + (3.68e8 + 6.12e8i)T$$
good2 $$1 + 48.0iT - 1.02e3T^{2}$$
5 $$1 + 1.21e3T + 9.76e6T^{2}$$
7 $$1 + 4.24e3T + 2.82e8T^{2}$$
11 $$1 + 3.39e4iT - 2.59e10T^{2}$$
13 $$1 + 4.13e5iT - 1.37e11T^{2}$$
17 $$1 + 2.03e6T + 2.01e12T^{2}$$
19 $$1 - 9.46e5T + 6.13e12T^{2}$$
23 $$1 - 2.44e6iT - 4.14e13T^{2}$$
29 $$1 + 1.24e7T + 4.20e14T^{2}$$
31 $$1 - 3.60e7iT - 8.19e14T^{2}$$
37 $$1 + 1.04e8iT - 4.80e15T^{2}$$
41 $$1 - 2.63e7T + 1.34e16T^{2}$$
43 $$1 - 1.45e8iT - 2.16e16T^{2}$$
47 $$1 - 2.23e8iT - 5.25e16T^{2}$$
53 $$1 + 3.03e8T + 1.74e17T^{2}$$
61 $$1 + 1.36e9iT - 7.13e17T^{2}$$
67 $$1 - 2.43e9iT - 1.82e18T^{2}$$
71 $$1 - 2.64e9T + 3.25e18T^{2}$$
73 $$1 + 8.53e8iT - 4.29e18T^{2}$$
79 $$1 - 4.49e7T + 9.46e18T^{2}$$
83 $$1 - 1.00e9iT - 1.55e19T^{2}$$
89 $$1 + 3.83e9iT - 3.11e19T^{2}$$
97 $$1 - 2.35e9iT - 7.37e19T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$