# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2}$ Sign $0.342 + 0.939i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.73i·3-s + (1.5 − 0.866i)5-s + (−1.5 − 0.866i)7-s − 2.99·9-s + (1.5 − 2.59i)11-s + (2.5 + 4.33i)13-s + (−1.49 − 2.59i)15-s − 6.92i·17-s + 3.46i·19-s + (−1.49 + 2.59i)21-s + (4.5 + 7.79i)23-s + (−1 + 1.73i)25-s + 5.19i·27-s + (1.5 + 0.866i)29-s + (−4.5 + 2.59i)31-s + ⋯
 L(s)  = 1 − 0.999i·3-s + (0.670 − 0.387i)5-s + (−0.566 − 0.327i)7-s − 0.999·9-s + (0.452 − 0.783i)11-s + (0.693 + 1.20i)13-s + (−0.387 − 0.670i)15-s − 1.68i·17-s + 0.794i·19-s + (−0.327 + 0.566i)21-s + (0.938 + 1.62i)23-s + (−0.200 + 0.346i)25-s + 0.999i·27-s + (0.278 + 0.160i)29-s + (−0.808 + 0.466i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$144$$    =    $$2^{4} \cdot 3^{2}$$ $$\varepsilon$$ = $0.342 + 0.939i$ motivic weight = $$1$$ character : $\chi_{144} (47, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 144,\ (\ :1/2),\ 0.342 + 0.939i)$$ $$L(1)$$ $$\approx$$ $$0.941878 - 0.659510i$$ $$L(\frac12)$$ $$\approx$$ $$0.941878 - 0.659510i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + 1.73iT$$
good5 $$1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2}$$
7 $$1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2}$$
11 $$1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + 6.92iT - 17T^{2}$$
19 $$1 - 3.46iT - 19T^{2}$$
23 $$1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (-4.5 - 2.59i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + 12T + 71T^{2}$$
73 $$1 + 2T + 73T^{2}$$
79 $$1 + (7.5 + 4.33i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 - 6.92iT - 89T^{2}$$
97 $$1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}