# Properties

 Degree $2$ Conductor $1344$ Sign $1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3·3-s − 20.3·5-s − 7·7-s + 9·9-s − 30.9·11-s − 50.6·13-s − 60.9·15-s − 102.·17-s − 61.2·19-s − 21·21-s − 148.·23-s + 287.·25-s + 27·27-s − 159.·29-s + 121.·31-s − 92.7·33-s + 142.·35-s + 357.·37-s − 151.·39-s + 466.·41-s − 185.·43-s − 182.·45-s + 131.·47-s + 49·49-s − 308.·51-s − 200.·53-s + 627.·55-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.81·5-s − 0.377·7-s + 0.333·9-s − 0.847·11-s − 1.07·13-s − 1.04·15-s − 1.46·17-s − 0.739·19-s − 0.218·21-s − 1.34·23-s + 2.29·25-s + 0.192·27-s − 1.01·29-s + 0.702·31-s − 0.489·33-s + 0.686·35-s + 1.59·37-s − 0.623·39-s + 1.77·41-s − 0.658·43-s − 0.605·45-s + 0.407·47-s + 0.142·49-s − 0.846·51-s − 0.518·53-s + 1.53·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ Sign: $1$ Motivic weight: $$3$$ Character: $\chi_{1344} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1344,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.4466545577$$ $$L(\frac12)$$ $$\approx$$ $$0.4466545577$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - 3T$$
7 $$1 + 7T$$
good5 $$1 + 20.3T + 125T^{2}$$
11 $$1 + 30.9T + 1.33e3T^{2}$$
13 $$1 + 50.6T + 2.19e3T^{2}$$
17 $$1 + 102.T + 4.91e3T^{2}$$
19 $$1 + 61.2T + 6.85e3T^{2}$$
23 $$1 + 148.T + 1.21e4T^{2}$$
29 $$1 + 159.T + 2.43e4T^{2}$$
31 $$1 - 121.T + 2.97e4T^{2}$$
37 $$1 - 357.T + 5.06e4T^{2}$$
41 $$1 - 466.T + 6.89e4T^{2}$$
43 $$1 + 185.T + 7.95e4T^{2}$$
47 $$1 - 131.T + 1.03e5T^{2}$$
53 $$1 + 200.T + 1.48e5T^{2}$$
59 $$1 + 591.T + 2.05e5T^{2}$$
61 $$1 + 70.5T + 2.26e5T^{2}$$
67 $$1 + 643.T + 3.00e5T^{2}$$
71 $$1 - 522.T + 3.57e5T^{2}$$
73 $$1 + 576.T + 3.89e5T^{2}$$
79 $$1 + 280.T + 4.93e5T^{2}$$
83 $$1 + 557.T + 5.71e5T^{2}$$
89 $$1 + 1.22e3T + 7.04e5T^{2}$$
97 $$1 - 65.0T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$