# Properties

 Label 2-1323-1.1-c3-0-63 Degree $2$ Conductor $1323$ Sign $1$ Analytic cond. $78.0595$ Root an. cond. $8.83513$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.09·2-s + 17.9·4-s − 18.5·5-s + 50.5·8-s − 94.7·10-s − 1.05·11-s − 58.7·13-s + 113.·16-s + 43.7·17-s + 131.·19-s − 333.·20-s − 5.37·22-s + 161.·23-s + 220.·25-s − 299.·26-s + 64.0·29-s + 55.9·31-s + 175.·32-s + 222.·34-s + 296.·37-s + 671.·38-s − 940.·40-s + 80.6·41-s − 134.·43-s − 18.9·44-s + 821.·46-s + 233.·47-s + ⋯
 L(s)  = 1 + 1.80·2-s + 2.24·4-s − 1.66·5-s + 2.23·8-s − 2.99·10-s − 0.0289·11-s − 1.25·13-s + 1.78·16-s + 0.624·17-s + 1.59·19-s − 3.72·20-s − 0.0521·22-s + 1.46·23-s + 1.76·25-s − 2.25·26-s + 0.409·29-s + 0.324·31-s + 0.971·32-s + 1.12·34-s + 1.31·37-s + 2.86·38-s − 3.71·40-s + 0.307·41-s − 0.476·43-s − 0.0648·44-s + 2.63·46-s + 0.724·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1323$$    =    $$3^{3} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$78.0595$$ Root analytic conductor: $$8.83513$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1323,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$5.151679682$$ $$L(\frac12)$$ $$\approx$$ $$5.151679682$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 - 5.09T + 8T^{2}$$
5 $$1 + 18.5T + 125T^{2}$$
11 $$1 + 1.05T + 1.33e3T^{2}$$
13 $$1 + 58.7T + 2.19e3T^{2}$$
17 $$1 - 43.7T + 4.91e3T^{2}$$
19 $$1 - 131.T + 6.85e3T^{2}$$
23 $$1 - 161.T + 1.21e4T^{2}$$
29 $$1 - 64.0T + 2.43e4T^{2}$$
31 $$1 - 55.9T + 2.97e4T^{2}$$
37 $$1 - 296.T + 5.06e4T^{2}$$
41 $$1 - 80.6T + 6.89e4T^{2}$$
43 $$1 + 134.T + 7.95e4T^{2}$$
47 $$1 - 233.T + 1.03e5T^{2}$$
53 $$1 - 387.T + 1.48e5T^{2}$$
59 $$1 - 722.T + 2.05e5T^{2}$$
61 $$1 + 388.T + 2.26e5T^{2}$$
67 $$1 + 730.T + 3.00e5T^{2}$$
71 $$1 + 685.T + 3.57e5T^{2}$$
73 $$1 - 275.T + 3.89e5T^{2}$$
79 $$1 - 854.T + 4.93e5T^{2}$$
83 $$1 - 922.T + 5.71e5T^{2}$$
89 $$1 + 301.T + 7.04e5T^{2}$$
97 $$1 + 913.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$