# Properties

 Degree 2 Conductor 3 Sign $0.555 - 0.831i$ Motivic weight 8 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 22.4i·2-s + (45 − 67.3i)3-s − 248·4-s − 224. i·5-s + (1.51e3 + 1.01e3i)6-s − 1.75e3·7-s + 179. i·8-s + (−2.51e3 − 6.06e3i)9-s + 5.04e3·10-s + 6.95e3i·11-s + (−1.11e4 + 1.67e4i)12-s + 2.57e4·13-s − 3.92e4i·14-s + (−1.51e4 − 1.01e4i)15-s − 6.75e4·16-s + 7.48e4i·17-s + ⋯
 L(s)  = 1 + 1.40i·2-s + (0.555 − 0.831i)3-s − 0.968·4-s − 0.359i·5-s + (1.16 + 0.779i)6-s − 0.728·7-s + 0.0438i·8-s + (−0.382 − 0.923i)9-s + 0.504·10-s + 0.475i·11-s + (−0.538 + 0.805i)12-s + 0.900·13-s − 1.02i·14-s + (−0.298 − 0.199i)15-s − 1.03·16-s + 0.896i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $0.555 - 0.831i$ motivic weight = $$8$$ character : $\chi_{3} (2, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3,\ (\ :4),\ 0.555 - 0.831i)$ $L(\frac{9}{2})$ $\approx$ $1.03711 + 0.554359i$ $L(\frac12)$ $\approx$ $1.03711 + 0.554359i$ $L(5)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 3$, $$F_p$$ is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + (-45 + 67.3i)T$$
good2 $$1 - 22.4iT - 256T^{2}$$
5 $$1 + 224. iT - 3.90e5T^{2}$$
7 $$1 + 1.75e3T + 5.76e6T^{2}$$
11 $$1 - 6.95e3iT - 2.14e8T^{2}$$
13 $$1 - 2.57e4T + 8.15e8T^{2}$$
17 $$1 - 7.48e4iT - 6.97e9T^{2}$$
19 $$1 - 1.89e4T + 1.69e10T^{2}$$
23 $$1 + 4.70e5iT - 7.83e10T^{2}$$
29 $$1 - 4.60e5iT - 5.00e11T^{2}$$
31 $$1 + 3.51e5T + 8.52e11T^{2}$$
37 $$1 - 1.33e6T + 3.51e12T^{2}$$
41 $$1 - 1.87e6iT - 7.98e12T^{2}$$
43 $$1 + 3.52e6T + 1.16e13T^{2}$$
47 $$1 + 4.08e6iT - 2.38e13T^{2}$$
53 $$1 + 6.60e6iT - 6.22e13T^{2}$$
59 $$1 - 1.37e7iT - 1.46e14T^{2}$$
61 $$1 - 7.53e5T + 1.91e14T^{2}$$
67 $$1 - 2.26e6T + 4.06e14T^{2}$$
71 $$1 - 1.70e7iT - 6.45e14T^{2}$$
73 $$1 - 2.76e7T + 8.06e14T^{2}$$
79 $$1 + 2.29e7T + 1.51e15T^{2}$$
83 $$1 + 4.63e7iT - 2.25e15T^{2}$$
89 $$1 - 7.26e7iT - 3.93e15T^{2}$$
97 $$1 - 1.47e8T + 7.83e15T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}