# Properties

 Label 2-165-1.1-c5-0-3 Degree $2$ Conductor $165$ Sign $1$ Analytic cond. $26.4633$ Root an. cond. $5.14425$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.30·2-s − 9·3-s − 13.4·4-s − 25·5-s − 38.7·6-s − 148.·7-s − 195.·8-s + 81·9-s − 107.·10-s + 121·11-s + 121.·12-s + 234.·13-s − 637.·14-s + 225·15-s − 409.·16-s + 1.03e3·17-s + 348.·18-s + 1.26e3·19-s + 337.·20-s + 1.33e3·21-s + 520.·22-s + 384.·23-s + 1.76e3·24-s + 625·25-s + 1.00e3·26-s − 729·27-s + 2.00e3·28-s + ⋯
 L(s)  = 1 + 0.760·2-s − 0.577·3-s − 0.421·4-s − 0.447·5-s − 0.438·6-s − 1.14·7-s − 1.08·8-s + 0.333·9-s − 0.340·10-s + 0.301·11-s + 0.243·12-s + 0.384·13-s − 0.869·14-s + 0.258·15-s − 0.400·16-s + 0.865·17-s + 0.253·18-s + 0.804·19-s + 0.188·20-s + 0.660·21-s + 0.229·22-s + 0.151·23-s + 0.624·24-s + 0.200·25-s + 0.292·26-s − 0.192·27-s + 0.482·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$165$$    =    $$3 \cdot 5 \cdot 11$$ Sign: $1$ Analytic conductor: $$26.4633$$ Root analytic conductor: $$5.14425$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 165,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.311744798$$ $$L(\frac12)$$ $$\approx$$ $$1.311744798$$ $$L(\frac{7}{2})$$ 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$$
5 $$1 + 25T$$
11 $$1 - 121T$$
good2 $$1 - 4.30T + 32T^{2}$$
7 $$1 + 148.T + 1.68e4T^{2}$$
13 $$1 - 234.T + 3.71e5T^{2}$$
17 $$1 - 1.03e3T + 1.41e6T^{2}$$
19 $$1 - 1.26e3T + 2.47e6T^{2}$$
23 $$1 - 384.T + 6.43e6T^{2}$$
29 $$1 + 4.48e3T + 2.05e7T^{2}$$
31 $$1 + 997.T + 2.86e7T^{2}$$
37 $$1 - 5.16e3T + 6.93e7T^{2}$$
41 $$1 - 2.25e3T + 1.15e8T^{2}$$
43 $$1 - 1.58e4T + 1.47e8T^{2}$$
47 $$1 - 1.20e4T + 2.29e8T^{2}$$
53 $$1 + 3.85e3T + 4.18e8T^{2}$$
59 $$1 + 2.02e4T + 7.14e8T^{2}$$
61 $$1 - 2.00e3T + 8.44e8T^{2}$$
67 $$1 - 4.09e4T + 1.35e9T^{2}$$
71 $$1 - 1.69e4T + 1.80e9T^{2}$$
73 $$1 + 5.66e4T + 2.07e9T^{2}$$
79 $$1 - 5.85e4T + 3.07e9T^{2}$$
83 $$1 + 5.22e4T + 3.93e9T^{2}$$
89 $$1 - 5.51e4T + 5.58e9T^{2}$$
97 $$1 - 9.93e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$