# Properties

 Degree $2$ Conductor $1344$ Sign $0.934 + 0.355i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3i·3-s − 15.6·5-s + (10.8 − 15.0i)7-s − 9·9-s − 41.3·11-s − 8.02·13-s + 47.0i·15-s + 14.2i·17-s + 10.2i·19-s + (−45.0 − 32.5i)21-s + 203. i·23-s + 121.·25-s + 27i·27-s − 82.6i·29-s − 156.·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s − 1.40·5-s + (0.585 − 0.810i)7-s − 0.333·9-s − 1.13·11-s − 0.171·13-s + 0.810i·15-s + 0.202i·17-s + 0.123i·19-s + (−0.468 − 0.337i)21-s + 1.84i·23-s + 0.969·25-s + 0.192i·27-s − 0.529i·29-s − 0.905·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ Sign: $0.934 + 0.355i$ Motivic weight: $$3$$ Character: $\chi_{1344} (223, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1344,\ (\ :3/2),\ 0.934 + 0.355i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.9447960393$$ $$L(\frac12)$$ $$\approx$$ $$0.9447960393$$ $$L(\frac{5}{2})$$ 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 + (-10.8 + 15.0i)T$$
good5 $$1 + 15.6T + 125T^{2}$$
11 $$1 + 41.3T + 1.33e3T^{2}$$
13 $$1 + 8.02T + 2.19e3T^{2}$$
17 $$1 - 14.2iT - 4.91e3T^{2}$$
19 $$1 - 10.2iT - 6.85e3T^{2}$$
23 $$1 - 203. iT - 1.21e4T^{2}$$
29 $$1 + 82.6iT - 2.43e4T^{2}$$
31 $$1 + 156.T + 2.97e4T^{2}$$
37 $$1 + 106. iT - 5.06e4T^{2}$$
41 $$1 - 315. iT - 6.89e4T^{2}$$
43 $$1 + 154.T + 7.95e4T^{2}$$
47 $$1 - 474.T + 1.03e5T^{2}$$
53 $$1 + 266. iT - 1.48e5T^{2}$$
59 $$1 - 425. iT - 2.05e5T^{2}$$
61 $$1 - 8.30T + 2.26e5T^{2}$$
67 $$1 + 820.T + 3.00e5T^{2}$$
71 $$1 + 109. iT - 3.57e5T^{2}$$
73 $$1 + 53.8iT - 3.89e5T^{2}$$
79 $$1 + 1.17e3iT - 4.93e5T^{2}$$
83 $$1 + 507. iT - 5.71e5T^{2}$$
89 $$1 - 1.06e3iT - 7.04e5T^{2}$$
97 $$1 + 573. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$