Properties

 Degree 2 Conductor $3 \cdot 7$ Sign $0.928 + 0.371i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (1.93 − 1.11i)2-s + (−2.93 + 0.614i)3-s + (0.5 − 0.866i)4-s + (−1.93 + 1.11i)5-s + (−5.00 + 4.47i)6-s + (3.5 − 6.06i)7-s + 6.70i·8-s + (8.24 − 3.60i)9-s + (−2.5 + 4.33i)10-s + (−9.68 − 5.59i)11-s + (−0.936 + 2.85i)12-s − 2·13-s − 15.6i·14-s + (5.00 − 4.47i)15-s + (9.5 + 16.4i)16-s + (23.2 + 13.4i)17-s + ⋯
 L(s)  = 1 + (0.968 − 0.559i)2-s + (−0.978 + 0.204i)3-s + (0.125 − 0.216i)4-s + (−0.387 + 0.223i)5-s + (−0.833 + 0.745i)6-s + (0.5 − 0.866i)7-s + 0.838i·8-s + (0.916 − 0.400i)9-s + (−0.250 + 0.433i)10-s + (−0.880 − 0.508i)11-s + (−0.0780 + 0.237i)12-s − 0.153·13-s − 1.11i·14-s + (0.333 − 0.298i)15-s + (0.593 + 1.02i)16-s + (1.36 + 0.789i)17-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$21$$    =    $$3 \cdot 7$$ $$\varepsilon$$ = $0.928 + 0.371i$ motivic weight = $$2$$ character : $\chi_{21} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 21,\ (\ :1),\ 0.928 + 0.371i)$ $L(\frac{3}{2})$ $\approx$ $0.992942 - 0.191397i$ $L(\frac12)$ $\approx$ $0.992942 - 0.191397i$ $L(2)$ 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 + (2.93 - 0.614i)T$$
7 $$1 + (-3.5 + 6.06i)T$$
good2 $$1 + (-1.93 + 1.11i)T + (2 - 3.46i)T^{2}$$
5 $$1 + (1.93 - 1.11i)T + (12.5 - 21.6i)T^{2}$$
11 $$1 + (9.68 + 5.59i)T + (60.5 + 104. i)T^{2}$$
13 $$1 + 2T + 169T^{2}$$
17 $$1 + (-23.2 - 13.4i)T + (144.5 + 250. i)T^{2}$$
19 $$1 + (8 + 13.8i)T + (-180.5 + 312. i)T^{2}$$
23 $$1 + (11.6 - 6.70i)T + (264.5 - 458. i)T^{2}$$
29 $$1 + 15.6iT - 841T^{2}$$
31 $$1 + (-1.5 + 2.59i)T + (-480.5 - 832. i)T^{2}$$
37 $$1 + (6 + 10.3i)T + (-684.5 + 1.18e3i)T^{2}$$
41 $$1 + 31.3iT - 1.68e3T^{2}$$
43 $$1 - 44T + 1.84e3T^{2}$$
47 $$1 + (-11.6 + 6.70i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (-17.4 - 10.0i)T + (1.40e3 + 2.43e3i)T^{2}$$
59 $$1 + (17.4 + 10.0i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (-13 - 22.5i)T + (-1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (26 - 45.0i)T + (-2.24e3 - 3.88e3i)T^{2}$$
71 $$1 + 93.9iT - 5.04e3T^{2}$$
73 $$1 + (9 - 15.5i)T + (-2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (-39.5 - 68.4i)T + (-3.12e3 + 5.40e3i)T^{2}$$
83 $$1 - 140. iT - 6.88e3T^{2}$$
89 $$1 + (42.6 - 24.5i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + 93T + 9.40e3T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}