# Properties

 Degree 2 Conductor $3 \cdot 7$ Sign $0.607 + 0.794i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.30i·2-s + (−1.82 − 2.38i)3-s + 2.29·4-s + 7.37i·5-s + (−3.11 + 2.38i)6-s − 2.64·7-s − 8.22i·8-s + (−2.35 + 8.68i)9-s + 9.64·10-s − 2.61i·11-s + (−4.17 − 5.45i)12-s − 6.35·13-s + 3.45i·14-s + (17.5 − 13.4i)15-s − 1.58·16-s − 12.1i·17-s + ⋯
 L(s)  = 1 − 0.653i·2-s + (−0.607 − 0.794i)3-s + 0.572·4-s + 1.47i·5-s + (−0.519 + 0.397i)6-s − 0.377·7-s − 1.02i·8-s + (−0.261 + 0.965i)9-s + 0.964·10-s − 0.237i·11-s + (−0.348 − 0.454i)12-s − 0.488·13-s + 0.247i·14-s + (1.17 − 0.896i)15-s − 0.0989·16-s − 0.714i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$21$$    =    $$3 \cdot 7$$ $$\varepsilon$$ = $0.607 + 0.794i$ motivic weight = $$2$$ character : $\chi_{21} (8, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 21,\ (\ :1),\ 0.607 + 0.794i)$ $L(\frac{3}{2})$ $\approx$ $0.742454 - 0.366798i$ $L(\frac12)$ $\approx$ $0.742454 - 0.366798i$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;7\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + (1.82 + 2.38i)T$$
7 $$1 + 2.64T$$
good2 $$1 + 1.30iT - 4T^{2}$$
5 $$1 - 7.37iT - 25T^{2}$$
11 $$1 + 2.61iT - 121T^{2}$$
13 $$1 + 6.35T + 169T^{2}$$
17 $$1 + 12.1iT - 289T^{2}$$
19 $$1 + 10.2T + 361T^{2}$$
23 $$1 - 4.30iT - 529T^{2}$$
29 $$1 - 17.3iT - 841T^{2}$$
31 $$1 - 39.2T + 961T^{2}$$
37 $$1 - 41.0T + 1.36e3T^{2}$$
41 $$1 + 30.2iT - 1.68e3T^{2}$$
43 $$1 + 55.8T + 1.84e3T^{2}$$
47 $$1 + 39.9iT - 2.20e3T^{2}$$
53 $$1 - 105. iT - 2.80e3T^{2}$$
59 $$1 - 41.3iT - 3.48e3T^{2}$$
61 $$1 + 20.4T + 3.72e3T^{2}$$
67 $$1 + 27.1T + 4.48e3T^{2}$$
71 $$1 + 67.8iT - 5.04e3T^{2}$$
73 $$1 - 60.7T + 5.32e3T^{2}$$
79 $$1 + 63.2T + 6.24e3T^{2}$$
83 $$1 - 89.9iT - 6.88e3T^{2}$$
89 $$1 - 63.1iT - 7.92e3T^{2}$$
97 $$1 - 19.1T + 9.40e3T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}