* June core orders +8.8 pct m/m vs +15.3 pct forecast * Core orders -10.4 pct in Q2, seen +2.9 pct in Q3 * Data suggests firms lack growth conviction * Capex the key to sustained economic recovery (Adds more details, context on trends) By Tetsushi Kajimoto TOKYO, Aug 14 (Reuters) - Japan's core machinery orders tumbled in April-June at their fastest since the last global financial crisis and only a modest rebound is seen in the current quarter - further challenging policymakers contending with a fragile economy.

The highly volatile data point, a key indicator of capital spending, followed news on Wednesday that the economy suffered its biggest contraction since 2011 in the second quarter as April's sales tax hike took a heavy toll on private consumption.

With exports and factory production weakening, policymakers had hoped business investment would drive a virtuous cycle of output, income generation and consumption, but Thursday's anaemic numbers cloud the outlook for sustained growth.

Cabinet Office data out on Thursday showed core orders fell 10.4 percent in April-June from the previous quarter, marking the first slide in five quarters and the sharpest drop since January-March 2009 when orders declined 12.3 percent.

Companies surveyed by the Cabinet Office forecast that core orders would rise 2.9 percent in July-September.

Orders at manufacturers and service-sector firms fell 8.5 percent and 6.7 percent in April-June respectively. Manufacturers see orders falling 0.5 percent in the current quarter, while service-sector firms expect a 2.2 percent gain.

Companies held off spending in April-June after boosting investment earlier this year, likely to meet demand related to upgrading Windows operating systems and tighter regulation on diesel vehicle emissions that kicked in from April, government officials said.

As the temporary factors run their course, analysts expect capital spending will be picking up from now on due to steady corporate earnings and the need for upgrades of ageing equipment, particularly among non-manufacturers, although the pace of recovery is likely to be moderate.

"The data suggests that any recovery in capital spending will be slack. What is lacking in Japan is companies' confidence in a growth outlook rather than inflation expectations," said Kyohei Morita, chief Japan economist at Barclays Capital.

"Japan must act quickly to implement its growth strategy, tackling issues such as cuts in the corporate tax rate, labour market reform and promotion of corporate governance." Core orders, which exclude ships and power generation gear, rose 8.8 percent in June from the prior month, well below a 15.3 percent gain forecast in a Reuters poll of economists, and following a record 19.5 percent drop in May.