# Properties

 Label 2-165-1.1-c5-0-29 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.17·2-s + 9·3-s − 5.19·4-s + 25·5-s + 46.5·6-s − 123.·7-s − 192.·8-s + 81·9-s + 129.·10-s − 121·11-s − 46.7·12-s − 500.·13-s − 639.·14-s + 225·15-s − 830.·16-s − 422.·17-s + 419.·18-s − 932.·19-s − 129.·20-s − 1.11e3·21-s − 626.·22-s − 1.22e3·23-s − 1.73e3·24-s + 625·25-s − 2.58e3·26-s + 729·27-s + 641.·28-s + ⋯
 L(s)  = 1 + 0.915·2-s + 0.577·3-s − 0.162·4-s + 0.447·5-s + 0.528·6-s − 0.952·7-s − 1.06·8-s + 0.333·9-s + 0.409·10-s − 0.301·11-s − 0.0937·12-s − 0.820·13-s − 0.871·14-s + 0.258·15-s − 0.811·16-s − 0.354·17-s + 0.305·18-s − 0.592·19-s − 0.0726·20-s − 0.549·21-s − 0.275·22-s − 0.482·23-s − 0.614·24-s + 0.200·25-s − 0.751·26-s + 0.192·27-s + 0.154·28-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: $$1$$ Selberg data: $$(2,\ 165,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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 - 5.17T + 32T^{2}$$
7 $$1 + 123.T + 1.68e4T^{2}$$
13 $$1 + 500.T + 3.71e5T^{2}$$
17 $$1 + 422.T + 1.41e6T^{2}$$
19 $$1 + 932.T + 2.47e6T^{2}$$
23 $$1 + 1.22e3T + 6.43e6T^{2}$$
29 $$1 + 2.11e3T + 2.05e7T^{2}$$
31 $$1 + 159.T + 2.86e7T^{2}$$
37 $$1 - 5.41e3T + 6.93e7T^{2}$$
41 $$1 + 1.80e4T + 1.15e8T^{2}$$
43 $$1 - 6.81e3T + 1.47e8T^{2}$$
47 $$1 + 1.50e4T + 2.29e8T^{2}$$
53 $$1 - 1.53e4T + 4.18e8T^{2}$$
59 $$1 - 2.34e4T + 7.14e8T^{2}$$
61 $$1 - 1.07e4T + 8.44e8T^{2}$$
67 $$1 - 1.45e4T + 1.35e9T^{2}$$
71 $$1 + 2.81e4T + 1.80e9T^{2}$$
73 $$1 + 2.88e4T + 2.07e9T^{2}$$
79 $$1 + 8.15e3T + 3.07e9T^{2}$$
83 $$1 + 1.09e5T + 3.93e9T^{2}$$
89 $$1 - 6.96e4T + 5.58e9T^{2}$$
97 $$1 - 9.15e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$