Properties

 Degree 2 Conductor $2^{2} \cdot 3^{4}$ Sign $0.276 - 0.961i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−3.19 + 2.40i)2-s + (4.42 − 15.3i)4-s − 2.03·5-s + 23.1i·7-s + (22.8 + 59.7i)8-s + (6.48 − 4.88i)10-s + 4.99i·11-s + 275.·13-s + (−55.6 − 73.9i)14-s + (−216. − 136. i)16-s − 266.·17-s − 367. i·19-s + (−8.98 + 31.2i)20-s + (−12.0 − 15.9i)22-s + 628. i·23-s + ⋯
 L(s)  = 1 + (−0.798 + 0.601i)2-s + (0.276 − 0.961i)4-s − 0.0812·5-s + 0.472i·7-s + (0.357 + 0.934i)8-s + (0.0648 − 0.0488i)10-s + 0.0413i·11-s + 1.63·13-s + (−0.283 − 0.377i)14-s + (−0.847 − 0.531i)16-s − 0.920·17-s − 1.01i·19-s + (−0.0224 + 0.0780i)20-s + (−0.0248 − 0.0330i)22-s + 1.18i·23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$324$$    =    $$2^{2} \cdot 3^{4}$$ $$\varepsilon$$ = $0.276 - 0.961i$ motivic weight = $$4$$ character : $\chi_{324} (163, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 324,\ (\ :2),\ 0.276 - 0.961i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$1.233182962$$ $$L(\frac12)$$ $$\approx$$ $$1.233182962$$ $$L(3)$$ 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 + (3.19 - 2.40i)T$$
3 $$1$$
good5 $$1 + 2.03T + 625T^{2}$$
7 $$1 - 23.1iT - 2.40e3T^{2}$$
11 $$1 - 4.99iT - 1.46e4T^{2}$$
13 $$1 - 275.T + 2.85e4T^{2}$$
17 $$1 + 266.T + 8.35e4T^{2}$$
19 $$1 + 367. iT - 1.30e5T^{2}$$
23 $$1 - 628. iT - 2.79e5T^{2}$$
29 $$1 - 638.T + 7.07e5T^{2}$$
31 $$1 + 1.37e3iT - 9.23e5T^{2}$$
37 $$1 - 1.46e3T + 1.87e6T^{2}$$
41 $$1 + 1.18e3T + 2.82e6T^{2}$$
43 $$1 - 1.65e3iT - 3.41e6T^{2}$$
47 $$1 - 355. iT - 4.87e6T^{2}$$
53 $$1 - 5.29e3T + 7.89e6T^{2}$$
59 $$1 - 6.03e3iT - 1.21e7T^{2}$$
61 $$1 - 1.66e3T + 1.38e7T^{2}$$
67 $$1 - 2.20e3iT - 2.01e7T^{2}$$
71 $$1 - 524. iT - 2.54e7T^{2}$$
73 $$1 + 1.49e3T + 2.83e7T^{2}$$
79 $$1 - 5.13e3iT - 3.89e7T^{2}$$
83 $$1 - 7.98e3iT - 4.74e7T^{2}$$
89 $$1 + 8.86e3T + 6.27e7T^{2}$$
97 $$1 - 6.81e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}