# Properties

 Label 2-165-1.1-c3-0-13 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 − 2.56·2-s + 3·3-s − 1.43·4-s − 5·5-s − 7.68·6-s + 6.24·7-s + 24.1·8-s + 9·9-s + 12.8·10-s − 11·11-s − 4.31·12-s − 49.1·13-s − 16·14-s − 15·15-s − 50.4·16-s + 82.7·17-s − 23.0·18-s − 130.·19-s + 7.19·20-s + 18.7·21-s + 28.1·22-s − 185.·23-s + 72.5·24-s + 25·25-s + 125.·26-s + 27·27-s − 8.98·28-s + ⋯
 L(s)  = 1 − 0.905·2-s + 0.577·3-s − 0.179·4-s − 0.447·5-s − 0.522·6-s + 0.337·7-s + 1.06·8-s + 0.333·9-s + 0.405·10-s − 0.301·11-s − 0.103·12-s − 1.04·13-s − 0.305·14-s − 0.258·15-s − 0.787·16-s + 1.17·17-s − 0.301·18-s − 1.57·19-s + 0.0804·20-s + 0.194·21-s + 0.273·22-s − 1.68·23-s + 0.616·24-s + 0.200·25-s + 0.949·26-s + 0.192·27-s − 0.0606·28-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 + 2.56T + 8T^{2}$$
7 $$1 - 6.24T + 343T^{2}$$
13 $$1 + 49.1T + 2.19e3T^{2}$$
17 $$1 - 82.7T + 4.91e3T^{2}$$
19 $$1 + 130.T + 6.85e3T^{2}$$
23 $$1 + 185.T + 1.21e4T^{2}$$
29 $$1 + 8.90T + 2.43e4T^{2}$$
31 $$1 - 5.26T + 2.97e4T^{2}$$
37 $$1 + 416.T + 5.06e4T^{2}$$
41 $$1 + 298.T + 6.89e4T^{2}$$
43 $$1 + 513.T + 7.95e4T^{2}$$
47 $$1 - 557.T + 1.03e5T^{2}$$
53 $$1 + 168.T + 1.48e5T^{2}$$
59 $$1 - 618.T + 2.05e5T^{2}$$
61 $$1 - 786.T + 2.26e5T^{2}$$
67 $$1 + 339.T + 3.00e5T^{2}$$
71 $$1 - 1.12e3T + 3.57e5T^{2}$$
73 $$1 + 123.T + 3.89e5T^{2}$$
79 $$1 + 309.T + 4.93e5T^{2}$$
83 $$1 + 1.02e3T + 5.71e5T^{2}$$
89 $$1 + 141.T + 7.04e5T^{2}$$
97 $$1 - 798.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$