# Properties

 Label 2-177-1.1-c7-0-67 Degree $2$ Conductor $177$ Sign $-1$ Analytic cond. $55.2921$ Root an. cond. $7.43586$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 22.0·2-s + 27·3-s + 357.·4-s − 492.·5-s + 594.·6-s − 1.20e3·7-s + 5.04e3·8-s + 729·9-s − 1.08e4·10-s − 7.75e3·11-s + 9.64e3·12-s + 1.92e3·13-s − 2.66e4·14-s − 1.32e4·15-s + 6.55e4·16-s − 2.18e4·17-s + 1.60e4·18-s − 3.16e4·19-s − 1.75e5·20-s − 3.26e4·21-s − 1.70e5·22-s + 7.60e4·23-s + 1.36e5·24-s + 1.64e5·25-s + 4.24e4·26-s + 1.96e4·27-s − 4.32e5·28-s + ⋯
 L(s)  = 1 + 1.94·2-s + 0.577·3-s + 2.79·4-s − 1.76·5-s + 1.12·6-s − 1.33·7-s + 3.48·8-s + 0.333·9-s − 3.43·10-s − 1.75·11-s + 1.61·12-s + 0.243·13-s − 2.59·14-s − 1.01·15-s + 3.99·16-s − 1.07·17-s + 0.649·18-s − 1.05·19-s − 4.91·20-s − 0.769·21-s − 3.42·22-s + 1.30·23-s + 2.01·24-s + 2.10·25-s + 0.473·26-s + 0.192·27-s − 3.72·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $-1$ Analytic conductor: $$55.2921$$ Root analytic conductor: $$7.43586$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: $\chi_{177} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 177,\ (\ :7/2),\ -1)$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 27T$$
59 $$1 + 2.05e5T$$
good2 $$1 - 22.0T + 128T^{2}$$
5 $$1 + 492.T + 7.81e4T^{2}$$
7 $$1 + 1.20e3T + 8.23e5T^{2}$$
11 $$1 + 7.75e3T + 1.94e7T^{2}$$
13 $$1 - 1.92e3T + 6.27e7T^{2}$$
17 $$1 + 2.18e4T + 4.10e8T^{2}$$
19 $$1 + 3.16e4T + 8.93e8T^{2}$$
23 $$1 - 7.60e4T + 3.40e9T^{2}$$
29 $$1 + 1.07e5T + 1.72e10T^{2}$$
31 $$1 + 5.68e4T + 2.75e10T^{2}$$
37 $$1 + 7.25e4T + 9.49e10T^{2}$$
41 $$1 + 5.98e5T + 1.94e11T^{2}$$
43 $$1 + 1.27e5T + 2.71e11T^{2}$$
47 $$1 - 2.07e4T + 5.06e11T^{2}$$
53 $$1 - 1.52e6T + 1.17e12T^{2}$$
61 $$1 - 2.40e6T + 3.14e12T^{2}$$
67 $$1 + 6.06e5T + 6.06e12T^{2}$$
71 $$1 + 3.52e6T + 9.09e12T^{2}$$
73 $$1 - 1.27e6T + 1.10e13T^{2}$$
79 $$1 - 3.89e6T + 1.92e13T^{2}$$
83 $$1 + 4.03e6T + 2.71e13T^{2}$$
89 $$1 - 5.97e6T + 4.42e13T^{2}$$
97 $$1 + 1.98e6T + 8.07e13T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$