# Properties

 Degree $2$ Conductor $1344$ Sign $0.820 + 0.571i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3i·3-s + 15.4·5-s + (−6.29 + 17.4i)7-s − 9·9-s + 16.8·11-s + 9.10·13-s + 46.2i·15-s − 73.9i·17-s − 151. i·19-s + (−52.2 − 18.8i)21-s − 0.264i·23-s + 112.·25-s − 27i·27-s − 279. i·29-s − 147.·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 1.37·5-s + (−0.339 + 0.940i)7-s − 0.333·9-s + 0.462·11-s + 0.194·13-s + 0.795i·15-s − 1.05i·17-s − 1.82i·19-s + (−0.543 − 0.196i)21-s − 0.00239i·23-s + 0.897·25-s − 0.192i·27-s − 1.79i·29-s − 0.856·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ Sign: $0.820 + 0.571i$ Motivic weight: $$3$$ Character: $\chi_{1344} (223, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1344,\ (\ :3/2),\ 0.820 + 0.571i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.401530264$$ $$L(\frac12)$$ $$\approx$$ $$2.401530264$$ $$L(\frac{5}{2})$$ 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 - 3iT$$
7 $$1 + (6.29 - 17.4i)T$$
good5 $$1 - 15.4T + 125T^{2}$$
11 $$1 - 16.8T + 1.33e3T^{2}$$
13 $$1 - 9.10T + 2.19e3T^{2}$$
17 $$1 + 73.9iT - 4.91e3T^{2}$$
19 $$1 + 151. iT - 6.85e3T^{2}$$
23 $$1 + 0.264iT - 1.21e4T^{2}$$
29 $$1 + 279. iT - 2.43e4T^{2}$$
31 $$1 + 147.T + 2.97e4T^{2}$$
37 $$1 + 418. iT - 5.06e4T^{2}$$
41 $$1 + 164. iT - 6.89e4T^{2}$$
43 $$1 - 266.T + 7.95e4T^{2}$$
47 $$1 - 277.T + 1.03e5T^{2}$$
53 $$1 - 276. iT - 1.48e5T^{2}$$
59 $$1 + 212. iT - 2.05e5T^{2}$$
61 $$1 - 174.T + 2.26e5T^{2}$$
67 $$1 + 317.T + 3.00e5T^{2}$$
71 $$1 - 106. iT - 3.57e5T^{2}$$
73 $$1 + 755. iT - 3.89e5T^{2}$$
79 $$1 - 194. iT - 4.93e5T^{2}$$
83 $$1 - 997. iT - 5.71e5T^{2}$$
89 $$1 - 988. iT - 7.04e5T^{2}$$
97 $$1 + 1.02e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$