Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 5$ Sign $0.419 + 0.907i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−23.4 − 50.7i)5-s − 10.2i·7-s + 596.·11-s + 420. i·13-s − 974. i·17-s − 380.·19-s − 3.54e3i·23-s + (−2.02e3 + 2.37e3i)25-s + 5.44e3·29-s + 3.62e3·31-s + (−520. + 240. i)35-s + 1.75e3i·37-s − 263.·41-s + 1.44e4i·43-s + 2.34e4i·47-s + ⋯
 L(s)  = 1 + (−0.419 − 0.907i)5-s − 0.0791i·7-s + 1.48·11-s + 0.690i·13-s − 0.817i·17-s − 0.241·19-s − 1.39i·23-s + (−0.648 + 0.760i)25-s + 1.20·29-s + 0.677·31-s + (−0.0718 + 0.0331i)35-s + 0.210i·37-s − 0.0245·41-s + 1.18i·43-s + 1.54i·47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$720$$    =    $$2^{4} \cdot 3^{2} \cdot 5$$ $$\varepsilon$$ = $0.419 + 0.907i$ motivic weight = $$5$$ character : $\chi_{720} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 720,\ (\ :5/2),\ 0.419 + 0.907i)$$ $$L(3)$$ $$\approx$$ $$2.180456982$$ $$L(\frac12)$$ $$\approx$$ $$2.180456982$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 + (23.4 + 50.7i)T$$
good7 $$1 + 10.2iT - 1.68e4T^{2}$$
11 $$1 - 596.T + 1.61e5T^{2}$$
13 $$1 - 420. iT - 3.71e5T^{2}$$
17 $$1 + 974. iT - 1.41e6T^{2}$$
19 $$1 + 380.T + 2.47e6T^{2}$$
23 $$1 + 3.54e3iT - 6.43e6T^{2}$$
29 $$1 - 5.44e3T + 2.05e7T^{2}$$
31 $$1 - 3.62e3T + 2.86e7T^{2}$$
37 $$1 - 1.75e3iT - 6.93e7T^{2}$$
41 $$1 + 263.T + 1.15e8T^{2}$$
43 $$1 - 1.44e4iT - 1.47e8T^{2}$$
47 $$1 - 2.34e4iT - 2.29e8T^{2}$$
53 $$1 - 3.34e4iT - 4.18e8T^{2}$$
59 $$1 - 2.90e3T + 7.14e8T^{2}$$
61 $$1 - 2.94e4T + 8.44e8T^{2}$$
67 $$1 + 7.16e3iT - 1.35e9T^{2}$$
71 $$1 + 8.13e4T + 1.80e9T^{2}$$
73 $$1 + 5.51e4iT - 2.07e9T^{2}$$
79 $$1 - 1.64e4T + 3.07e9T^{2}$$
83 $$1 + 1.16e5iT - 3.93e9T^{2}$$
89 $$1 - 9.93e4T + 5.58e9T^{2}$$
97 $$1 - 6.29e4iT - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}