# Properties

 Degree $2$ Conductor $1344$ Sign $0.921 - 0.387i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.22 − 1.22i)3-s − 2.44·5-s + (1 − 2.44i)7-s + 2.99i·9-s + 2.44i·13-s + (2.99 + 2.99i)15-s − 4.89·17-s + 2.44i·19-s + (−4.22 + 1.77i)21-s + 6i·23-s + 0.999·25-s + (3.67 − 3.67i)27-s − 6i·29-s + (−2.44 + 5.99i)35-s + 2·37-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)3-s − 1.09·5-s + (0.377 − 0.925i)7-s + 0.999i·9-s + 0.679i·13-s + (0.774 + 0.774i)15-s − 1.18·17-s + 0.561i·19-s + (−0.921 + 0.387i)21-s + 1.25i·23-s + 0.199·25-s + (0.707 − 0.707i)27-s − 1.11i·29-s + (−0.414 + 1.01i)35-s + 0.328·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ Sign: $0.921 - 0.387i$ Motivic weight: $$1$$ Character: $\chi_{1344} (1217, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1344,\ (\ :1/2),\ 0.921 - 0.387i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7504822354$$ $$L(\frac12)$$ $$\approx$$ $$0.7504822354$$ $$L(\frac{3}{2})$$ 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 + (1.22 + 1.22i)T$$
7 $$1 + (-1 + 2.44i)T$$
good5 $$1 + 2.44T + 5T^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 - 2.44iT - 13T^{2}$$
17 $$1 + 4.89T + 17T^{2}$$
19 $$1 - 2.44iT - 19T^{2}$$
23 $$1 - 6iT - 23T^{2}$$
29 $$1 + 6iT - 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 - 4.89T + 41T^{2}$$
43 $$1 - 4T + 43T^{2}$$
47 $$1 - 4.89T + 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 - 12.2T + 59T^{2}$$
61 $$1 - 12.2iT - 61T^{2}$$
67 $$1 - 8T + 67T^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 - 9.79iT - 73T^{2}$$
79 $$1 - 10T + 79T^{2}$$
83 $$1 + 2.44T + 83T^{2}$$
89 $$1 + 89T^{2}$$
97 $$1 - 4.89iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$