Properties

 Degree 2 Conductor $2^{4} \cdot 3$ Sign $0.857 - 0.513i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (1.87 + 0.697i)2-s + (1.22 + 1.22i)3-s + (3.02 + 2.61i)4-s + (−5.24 − 5.24i)5-s + (1.44 + 3.14i)6-s − 5.32·7-s + (3.85 + 7.01i)8-s + 2.99i·9-s + (−6.17 − 13.4i)10-s + (12.2 − 12.2i)11-s + (0.507 + 6.90i)12-s + (−5.73 + 5.73i)13-s + (−9.98 − 3.71i)14-s − 12.8i·15-s + (2.33 + 15.8i)16-s − 23.3·17-s + ⋯
 L(s)  = 1 + (0.937 + 0.348i)2-s + (0.408 + 0.408i)3-s + (0.757 + 0.653i)4-s + (−1.04 − 1.04i)5-s + (0.240 + 0.524i)6-s − 0.761·7-s + (0.481 + 0.876i)8-s + 0.333i·9-s + (−0.617 − 1.34i)10-s + (1.11 − 1.11i)11-s + (0.0423 + 0.575i)12-s + (−0.441 + 0.441i)13-s + (−0.713 − 0.265i)14-s − 0.856i·15-s + (0.146 + 0.989i)16-s − 1.37·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$48$$    =    $$2^{4} \cdot 3$$ $$\varepsilon$$ = $0.857 - 0.513i$ motivic weight = $$2$$ character : $\chi_{48} (43, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 48,\ (\ :1),\ 0.857 - 0.513i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.65617 + 0.457865i$$ $$L(\frac12)$$ $$\approx$$ $$1.65617 + 0.457865i$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ 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 + (-1.87 - 0.697i)T$$
3 $$1 + (-1.22 - 1.22i)T$$
good5 $$1 + (5.24 + 5.24i)T + 25iT^{2}$$
7 $$1 + 5.32T + 49T^{2}$$
11 $$1 + (-12.2 + 12.2i)T - 121iT^{2}$$
13 $$1 + (5.73 - 5.73i)T - 169iT^{2}$$
17 $$1 + 23.3T + 289T^{2}$$
19 $$1 + (-11.7 - 11.7i)T + 361iT^{2}$$
23 $$1 - 5.80T + 529T^{2}$$
29 $$1 + (-18.3 + 18.3i)T - 841iT^{2}$$
31 $$1 - 16.9iT - 961T^{2}$$
37 $$1 + (-15.3 - 15.3i)T + 1.36e3iT^{2}$$
41 $$1 - 29.2iT - 1.68e3T^{2}$$
43 $$1 + (-33.4 + 33.4i)T - 1.84e3iT^{2}$$
47 $$1 - 18.2iT - 2.20e3T^{2}$$
53 $$1 + (66.9 + 66.9i)T + 2.80e3iT^{2}$$
59 $$1 + (27.1 - 27.1i)T - 3.48e3iT^{2}$$
61 $$1 + (-65.2 + 65.2i)T - 3.72e3iT^{2}$$
67 $$1 + (37.6 + 37.6i)T + 4.48e3iT^{2}$$
71 $$1 - 42.6T + 5.04e3T^{2}$$
73 $$1 + 106. iT - 5.32e3T^{2}$$
79 $$1 - 21.2iT - 6.24e3T^{2}$$
83 $$1 + (-24.1 - 24.1i)T + 6.88e3iT^{2}$$
89 $$1 + 52.8iT - 7.92e3T^{2}$$
97 $$1 + 21.0T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}