# Properties

 Label 2-177-1.1-c5-0-34 Degree $2$ Conductor $177$ Sign $-1$ Analytic cond. $28.3879$ Root an. cond. $5.32803$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.65·2-s − 9·3-s − 29.2·4-s + 59.9·5-s − 14.9·6-s − 87.2·7-s − 101.·8-s + 81·9-s + 99.4·10-s + 655.·11-s + 263.·12-s + 139.·13-s − 144.·14-s − 539.·15-s + 767.·16-s − 1.51e3·17-s + 134.·18-s + 393.·19-s − 1.75e3·20-s + 785.·21-s + 1.08e3·22-s − 4.74e3·23-s + 914.·24-s + 466.·25-s + 232.·26-s − 729·27-s + 2.55e3·28-s + ⋯
 L(s)  = 1 + 0.293·2-s − 0.577·3-s − 0.913·4-s + 1.07·5-s − 0.169·6-s − 0.672·7-s − 0.561·8-s + 0.333·9-s + 0.314·10-s + 1.63·11-s + 0.527·12-s + 0.229·13-s − 0.197·14-s − 0.618·15-s + 0.749·16-s − 1.27·17-s + 0.0977·18-s + 0.250·19-s − 0.979·20-s + 0.388·21-s + 0.479·22-s − 1.87·23-s + 0.324·24-s + 0.149·25-s + 0.0673·26-s − 0.192·27-s + 0.615·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $-1$ Analytic conductor: $$28.3879$$ Root analytic conductor: $$5.32803$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{177} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 177,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 9T$$
59 $$1 + 3.48e3T$$
good2 $$1 - 1.65T + 32T^{2}$$
5 $$1 - 59.9T + 3.12e3T^{2}$$
7 $$1 + 87.2T + 1.68e4T^{2}$$
11 $$1 - 655.T + 1.61e5T^{2}$$
13 $$1 - 139.T + 3.71e5T^{2}$$
17 $$1 + 1.51e3T + 1.41e6T^{2}$$
19 $$1 - 393.T + 2.47e6T^{2}$$
23 $$1 + 4.74e3T + 6.43e6T^{2}$$
29 $$1 - 2.79e3T + 2.05e7T^{2}$$
31 $$1 + 1.80e3T + 2.86e7T^{2}$$
37 $$1 - 6.71e3T + 6.93e7T^{2}$$
41 $$1 + 1.39e4T + 1.15e8T^{2}$$
43 $$1 + 1.93e4T + 1.47e8T^{2}$$
47 $$1 + 1.57e4T + 2.29e8T^{2}$$
53 $$1 + 2.56e4T + 4.18e8T^{2}$$
61 $$1 - 1.10e4T + 8.44e8T^{2}$$
67 $$1 + 3.45e4T + 1.35e9T^{2}$$
71 $$1 - 4.30e4T + 1.80e9T^{2}$$
73 $$1 + 3.49e4T + 2.07e9T^{2}$$
79 $$1 - 9.11e4T + 3.07e9T^{2}$$
83 $$1 + 1.09e5T + 3.93e9T^{2}$$
89 $$1 - 1.36e5T + 5.58e9T^{2}$$
97 $$1 + 1.39e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$