# Properties

 Degree 2 Conductor 3 Sign $0.662 - 0.749i$ Motivic weight 16 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 198. i·2-s + (4.34e3 − 4.91e3i)3-s + 2.60e4·4-s + 4.82e5i·5-s + (9.77e5 + 8.63e5i)6-s + 5.53e6·7-s + 1.81e7i·8-s + (−5.31e6 − 4.27e7i)9-s − 9.58e7·10-s − 1.03e8i·11-s + (1.13e8 − 1.27e8i)12-s − 7.54e7·13-s + 1.10e9i·14-s + (2.37e9 + 2.09e9i)15-s − 1.91e9·16-s − 7.90e9i·17-s + ⋯
 L(s)  = 1 + 0.776i·2-s + (0.662 − 0.749i)3-s + 0.397·4-s + 1.23i·5-s + (0.581 + 0.514i)6-s + 0.960·7-s + 1.08i·8-s + (−0.123 − 0.992i)9-s − 0.958·10-s − 0.484i·11-s + (0.262 − 0.297i)12-s − 0.0925·13-s + 0.745i·14-s + (0.925 + 0.817i)15-s − 0.445·16-s − 1.13i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(17-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $0.662 - 0.749i$ motivic weight = $$16$$ character : $\chi_{3} (2, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3,\ (\ :8),\ 0.662 - 0.749i)$ $L(\frac{17}{2})$ $\approx$ $1.96194 + 0.884693i$ $L(\frac12)$ $\approx$ $1.96194 + 0.884693i$ $L(9)$ 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 + (-4.34e3 + 4.91e3i)T$$
good2 $$1 - 198. iT - 6.55e4T^{2}$$
5 $$1 - 4.82e5iT - 1.52e11T^{2}$$
7 $$1 - 5.53e6T + 3.32e13T^{2}$$
11 $$1 + 1.03e8iT - 4.59e16T^{2}$$
13 $$1 + 7.54e7T + 6.65e17T^{2}$$
17 $$1 + 7.90e9iT - 4.86e19T^{2}$$
19 $$1 + 3.27e10T + 2.88e20T^{2}$$
23 $$1 - 4.02e10iT - 6.13e21T^{2}$$
29 $$1 + 6.12e11iT - 2.50e23T^{2}$$
31 $$1 - 5.76e11T + 7.27e23T^{2}$$
37 $$1 + 2.03e11T + 1.23e25T^{2}$$
41 $$1 - 2.51e12iT - 6.37e25T^{2}$$
43 $$1 - 2.81e12T + 1.36e26T^{2}$$
47 $$1 - 1.80e13iT - 5.66e26T^{2}$$
53 $$1 + 2.59e13iT - 3.87e27T^{2}$$
59 $$1 + 1.63e14iT - 2.15e28T^{2}$$
61 $$1 - 7.04e13T + 3.67e28T^{2}$$
67 $$1 + 1.75e14T + 1.64e29T^{2}$$
71 $$1 - 6.72e14iT - 4.16e29T^{2}$$
73 $$1 - 1.04e15T + 6.50e29T^{2}$$
79 $$1 - 4.55e13T + 2.30e30T^{2}$$
83 $$1 + 3.65e14iT - 5.07e30T^{2}$$
89 $$1 - 6.68e15iT - 1.54e31T^{2}$$
97 $$1 + 5.40e15T + 6.14e31T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}