# Properties

 Label 2-165-1.1-c3-0-16 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 + 5.17·2-s + 3·3-s + 18.7·4-s + 5·5-s + 15.5·6-s − 11.1·7-s + 55.5·8-s + 9·9-s + 25.8·10-s − 11·11-s + 56.2·12-s − 89.5·13-s − 57.7·14-s + 15·15-s + 137.·16-s + 58.3·17-s + 46.5·18-s + 24.5·19-s + 93.6·20-s − 33.5·21-s − 56.8·22-s − 111.·23-s + 166.·24-s + 25·25-s − 462.·26-s + 27·27-s − 209.·28-s + ⋯
 L(s)  = 1 + 1.82·2-s + 0.577·3-s + 2.34·4-s + 0.447·5-s + 1.05·6-s − 0.603·7-s + 2.45·8-s + 0.333·9-s + 0.817·10-s − 0.301·11-s + 1.35·12-s − 1.91·13-s − 1.10·14-s + 0.258·15-s + 2.14·16-s + 0.832·17-s + 0.609·18-s + 0.296·19-s + 1.04·20-s − 0.348·21-s − 0.551·22-s − 1.01·23-s + 1.41·24-s + 0.200·25-s − 3.49·26-s + 0.192·27-s − 1.41·28-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

 $$L(2)$$ $$\approx$$ $$5.546198562$$ $$L(\frac12)$$ $$\approx$$ $$5.546198562$$ $$L(\frac{5}{2})$$ 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 - 5.17T + 8T^{2}$$
7 $$1 + 11.1T + 343T^{2}$$
13 $$1 + 89.5T + 2.19e3T^{2}$$
17 $$1 - 58.3T + 4.91e3T^{2}$$
19 $$1 - 24.5T + 6.85e3T^{2}$$
23 $$1 + 111.T + 1.21e4T^{2}$$
29 $$1 - 109.T + 2.43e4T^{2}$$
31 $$1 - 119.T + 2.97e4T^{2}$$
37 $$1 + 356.T + 5.06e4T^{2}$$
41 $$1 - 268.T + 6.89e4T^{2}$$
43 $$1 - 263.T + 7.95e4T^{2}$$
47 $$1 + 206.T + 1.03e5T^{2}$$
53 $$1 - 223.T + 1.48e5T^{2}$$
59 $$1 - 475.T + 2.05e5T^{2}$$
61 $$1 + 513.T + 2.26e5T^{2}$$
67 $$1 + 264.T + 3.00e5T^{2}$$
71 $$1 + 1.11e3T + 3.57e5T^{2}$$
73 $$1 - 893.T + 3.89e5T^{2}$$
79 $$1 - 1.30e3T + 4.93e5T^{2}$$
83 $$1 - 1.04e3T + 5.71e5T^{2}$$
89 $$1 + 1.41e3T + 7.04e5T^{2}$$
97 $$1 - 85.8T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$