Properties

 Degree 2 Conductor $2^{2} \cdot 3^{2} \cdot 7^{2}$ Sign $0.999 - 0.0131i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−0.658 − 1.25i)2-s + (−1.13 + 1.64i)4-s + 2.08i·5-s + (2.80 + 0.333i)8-s + (2.60 − 1.37i)10-s + 4.26·11-s + 4.80·13-s + (−1.43 − 3.73i)16-s − 3.20i·17-s + 2.81i·19-s + (−3.42 − 2.35i)20-s + (−2.80 − 5.34i)22-s − 4.66·23-s + 0.669·25-s + (−3.16 − 6.01i)26-s + ⋯
 L(s)  = 1 + (−0.465 − 0.885i)2-s + (−0.566 + 0.823i)4-s + 0.930i·5-s + (0.993 + 0.117i)8-s + (0.823 − 0.433i)10-s + 1.28·11-s + 1.33·13-s + (−0.357 − 0.933i)16-s − 0.777i·17-s + 0.646i·19-s + (−0.766 − 0.527i)20-s + (−0.599 − 1.13i)22-s − 0.971·23-s + 0.133·25-s + (−0.620 − 1.17i)26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $0.999 - 0.0131i$ motivic weight = $$1$$ character : $\chi_{1764} (1079, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1764,\ (\ :1/2),\ 0.999 - 0.0131i)$$ $$L(1)$$ $$\approx$$ $$1.442491901$$ $$L(\frac12)$$ $$\approx$$ $$1.442491901$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (0.658 + 1.25i)T$$
3 $$1$$
7 $$1$$
good5 $$1 - 2.08iT - 5T^{2}$$
11 $$1 - 4.26T + 11T^{2}$$
13 $$1 - 4.80T + 13T^{2}$$
17 $$1 + 3.20iT - 17T^{2}$$
19 $$1 - 2.81iT - 19T^{2}$$
23 $$1 + 4.66T + 23T^{2}$$
29 $$1 + 3.87iT - 29T^{2}$$
31 $$1 - 10.2iT - 31T^{2}$$
37 $$1 - 0.273T + 37T^{2}$$
41 $$1 - 0.387iT - 41T^{2}$$
43 $$1 - 0.907iT - 43T^{2}$$
47 $$1 + 7.85T + 47T^{2}$$
53 $$1 + 11.7iT - 53T^{2}$$
59 $$1 - 3.70T + 59T^{2}$$
61 $$1 - 8.02T + 61T^{2}$$
67 $$1 - 1.40iT - 67T^{2}$$
71 $$1 - 11.9T + 71T^{2}$$
73 $$1 - 12.2T + 73T^{2}$$
79 $$1 - 0.826iT - 79T^{2}$$
83 $$1 - 5.69T + 83T^{2}$$
89 $$1 - 3.02iT - 89T^{2}$$
97 $$1 - 15.2T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}