# Properties

 Label 2-165-1.1-c3-0-3 Degree $2$ Conductor $165$ Sign $1$ Analytic cond. $9.73531$ Root an. cond. $3.12014$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.793·2-s − 3·3-s − 7.37·4-s + 5·5-s − 2.38·6-s − 2.90·7-s − 12.1·8-s + 9·9-s + 3.96·10-s + 11·11-s + 22.1·12-s + 68.4·13-s − 2.30·14-s − 15·15-s + 49.2·16-s − 31.0·17-s + 7.14·18-s + 54.9·19-s − 36.8·20-s + 8.72·21-s + 8.72·22-s + 180.·23-s + 36.5·24-s + 25·25-s + 54.3·26-s − 27·27-s + 21.4·28-s + ⋯
 L(s)  = 1 + 0.280·2-s − 0.577·3-s − 0.921·4-s + 0.447·5-s − 0.161·6-s − 0.157·7-s − 0.539·8-s + 0.333·9-s + 0.125·10-s + 0.301·11-s + 0.531·12-s + 1.46·13-s − 0.0440·14-s − 0.258·15-s + 0.770·16-s − 0.443·17-s + 0.0935·18-s + 0.663·19-s − 0.412·20-s + 0.0906·21-s + 0.0845·22-s + 1.63·23-s + 0.311·24-s + 0.200·25-s + 0.409·26-s − 0.192·27-s + 0.144·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$165$$    =    $$3 \cdot 5 \cdot 11$$ Sign: $1$ Analytic conductor: $$9.73531$$ Root analytic conductor: $$3.12014$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 165,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.434502568$$ $$L(\frac12)$$ $$\approx$$ $$1.434502568$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 3T$$
5 $$1 - 5T$$
11 $$1 - 11T$$
good2 $$1 - 0.793T + 8T^{2}$$
7 $$1 + 2.90T + 343T^{2}$$
13 $$1 - 68.4T + 2.19e3T^{2}$$
17 $$1 + 31.0T + 4.91e3T^{2}$$
19 $$1 - 54.9T + 6.85e3T^{2}$$
23 $$1 - 180.T + 1.21e4T^{2}$$
29 $$1 - 67.3T + 2.43e4T^{2}$$
31 $$1 - 153.T + 2.97e4T^{2}$$
37 $$1 + 324.T + 5.06e4T^{2}$$
41 $$1 + 25.4T + 6.89e4T^{2}$$
43 $$1 - 133.T + 7.95e4T^{2}$$
47 $$1 - 113.T + 1.03e5T^{2}$$
53 $$1 - 91.6T + 1.48e5T^{2}$$
59 $$1 - 434.T + 2.05e5T^{2}$$
61 $$1 + 60.2T + 2.26e5T^{2}$$
67 $$1 + 439.T + 3.00e5T^{2}$$
71 $$1 - 436.T + 3.57e5T^{2}$$
73 $$1 + 91.5T + 3.89e5T^{2}$$
79 $$1 - 947.T + 4.93e5T^{2}$$
83 $$1 + 944.T + 5.71e5T^{2}$$
89 $$1 - 413.T + 7.04e5T^{2}$$
97 $$1 + 1.46e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$