Properties

 Degree $2$ Conductor $1344$ Sign $-0.943 - 0.332i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 3i·3-s − 1.64·5-s + (15.2 + 10.4i)7-s − 9·9-s + 20.3·11-s − 13.0·13-s − 4.92i·15-s − 23.9i·17-s + 87.7i·19-s + (−31.4 + 45.8i)21-s + 73.6i·23-s − 122.·25-s − 27i·27-s − 58.9i·29-s − 124.·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s − 0.146·5-s + (0.824 + 0.565i)7-s − 0.333·9-s + 0.557·11-s − 0.279·13-s − 0.0847i·15-s − 0.341i·17-s + 1.05i·19-s + (−0.326 + 0.476i)21-s + 0.667i·23-s − 0.978·25-s − 0.192i·27-s − 0.377i·29-s − 0.723·31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ Sign: $-0.943 - 0.332i$ Motivic weight: $$3$$ Character: $\chi_{1344} (223, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1344,\ (\ :3/2),\ -0.943 - 0.332i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.323172908$$ $$L(\frac12)$$ $$\approx$$ $$1.323172908$$ $$L(\frac{5}{2})$$ 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 + (-15.2 - 10.4i)T$$
good5 $$1 + 1.64T + 125T^{2}$$
11 $$1 - 20.3T + 1.33e3T^{2}$$
13 $$1 + 13.0T + 2.19e3T^{2}$$
17 $$1 + 23.9iT - 4.91e3T^{2}$$
19 $$1 - 87.7iT - 6.85e3T^{2}$$
23 $$1 - 73.6iT - 1.21e4T^{2}$$
29 $$1 + 58.9iT - 2.43e4T^{2}$$
31 $$1 + 124.T + 2.97e4T^{2}$$
37 $$1 - 56.5iT - 5.06e4T^{2}$$
41 $$1 + 135. iT - 6.89e4T^{2}$$
43 $$1 - 259.T + 7.95e4T^{2}$$
47 $$1 - 217.T + 1.03e5T^{2}$$
53 $$1 - 529. iT - 1.48e5T^{2}$$
59 $$1 - 685. iT - 2.05e5T^{2}$$
61 $$1 + 149.T + 2.26e5T^{2}$$
67 $$1 + 409.T + 3.00e5T^{2}$$
71 $$1 + 885. iT - 3.57e5T^{2}$$
73 $$1 - 269. iT - 3.89e5T^{2}$$
79 $$1 + 902. iT - 4.93e5T^{2}$$
83 $$1 - 623. iT - 5.71e5T^{2}$$
89 $$1 - 986. iT - 7.04e5T^{2}$$
97 $$1 - 179. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$