# Properties

 Label 2-165-1.1-c3-0-17 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.56·2-s + 3·3-s − 5.56·4-s − 5·5-s + 4.68·6-s − 10.2·7-s − 21.1·8-s + 9·9-s − 7.80·10-s − 11·11-s − 16.6·12-s − 40.8·13-s − 16·14-s − 15·15-s + 11.4·16-s − 98.7·17-s + 14.0·18-s − 39.6·19-s + 27.8·20-s − 30.7·21-s − 17.1·22-s + 61.6·23-s − 63.5·24-s + 25·25-s − 63.8·26-s + 27·27-s + 56.9·28-s + ⋯
 L(s)  = 1 + 0.552·2-s + 0.577·3-s − 0.695·4-s − 0.447·5-s + 0.318·6-s − 0.553·7-s − 0.935·8-s + 0.333·9-s − 0.246·10-s − 0.301·11-s − 0.401·12-s − 0.872·13-s − 0.305·14-s − 0.258·15-s + 0.178·16-s − 1.40·17-s + 0.184·18-s − 0.478·19-s + 0.310·20-s − 0.319·21-s − 0.166·22-s + 0.559·23-s − 0.540·24-s + 0.200·25-s − 0.481·26-s + 0.192·27-s + 0.384·28-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: $$1$$ Selberg data: $$(2,\ 165,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 1.56T + 8T^{2}$$
7 $$1 + 10.2T + 343T^{2}$$
13 $$1 + 40.8T + 2.19e3T^{2}$$
17 $$1 + 98.7T + 4.91e3T^{2}$$
19 $$1 + 39.6T + 6.85e3T^{2}$$
23 $$1 - 61.6T + 1.21e4T^{2}$$
29 $$1 + 149.T + 2.43e4T^{2}$$
31 $$1 - 54.7T + 2.97e4T^{2}$$
37 $$1 - 44.8T + 5.06e4T^{2}$$
41 $$1 - 336.T + 6.89e4T^{2}$$
43 $$1 + 2.36T + 7.95e4T^{2}$$
47 $$1 + 333.T + 1.03e5T^{2}$$
53 $$1 - 640.T + 1.48e5T^{2}$$
59 $$1 + 370.T + 2.05e5T^{2}$$
61 $$1 + 714.T + 2.26e5T^{2}$$
67 $$1 + 404.T + 3.00e5T^{2}$$
71 $$1 - 939.T + 3.57e5T^{2}$$
73 $$1 + 362.T + 3.89e5T^{2}$$
79 $$1 - 951.T + 4.93e5T^{2}$$
83 $$1 - 735.T + 5.71e5T^{2}$$
89 $$1 - 385.T + 7.04e5T^{2}$$
97 $$1 + 966.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$