# Properties

 Label 2-165-1.1-c5-0-31 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 + 7.52·2-s + 9·3-s + 24.6·4-s − 25·5-s + 67.7·6-s − 234.·7-s − 55.3·8-s + 81·9-s − 188.·10-s + 121·11-s + 221.·12-s + 236.·13-s − 1.76e3·14-s − 225·15-s − 1.20e3·16-s − 608.·17-s + 609.·18-s − 1.79e3·19-s − 616.·20-s − 2.10e3·21-s + 910.·22-s − 4.77e3·23-s − 498.·24-s + 625·25-s + 1.77e3·26-s + 729·27-s − 5.76e3·28-s + ⋯
 L(s)  = 1 + 1.33·2-s + 0.577·3-s + 0.770·4-s − 0.447·5-s + 0.768·6-s − 1.80·7-s − 0.305·8-s + 0.333·9-s − 0.594·10-s + 0.301·11-s + 0.444·12-s + 0.387·13-s − 2.40·14-s − 0.258·15-s − 1.17·16-s − 0.510·17-s + 0.443·18-s − 1.14·19-s − 0.344·20-s − 1.04·21-s + 0.401·22-s − 1.88·23-s − 0.176·24-s + 0.200·25-s + 0.515·26-s + 0.192·27-s − 1.39·28-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 - 7.52T + 32T^{2}$$
7 $$1 + 234.T + 1.68e4T^{2}$$
13 $$1 - 236.T + 3.71e5T^{2}$$
17 $$1 + 608.T + 1.41e6T^{2}$$
19 $$1 + 1.79e3T + 2.47e6T^{2}$$
23 $$1 + 4.77e3T + 6.43e6T^{2}$$
29 $$1 - 2.80e3T + 2.05e7T^{2}$$
31 $$1 - 1.02e4T + 2.86e7T^{2}$$
37 $$1 + 7.62e3T + 6.93e7T^{2}$$
41 $$1 - 8.35e3T + 1.15e8T^{2}$$
43 $$1 + 1.19e4T + 1.47e8T^{2}$$
47 $$1 - 8.38e3T + 2.29e8T^{2}$$
53 $$1 - 3.45e3T + 4.18e8T^{2}$$
59 $$1 + 5.10e4T + 7.14e8T^{2}$$
61 $$1 - 9.98e3T + 8.44e8T^{2}$$
67 $$1 + 3.65e4T + 1.35e9T^{2}$$
71 $$1 + 3.05e4T + 1.80e9T^{2}$$
73 $$1 - 8.38e4T + 2.07e9T^{2}$$
79 $$1 - 1.03e5T + 3.07e9T^{2}$$
83 $$1 + 4.34e3T + 3.93e9T^{2}$$
89 $$1 + 4.54e4T + 5.58e9T^{2}$$
97 $$1 + 1.83e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$