I am trying to use apex:actionFunction that has one parameter in my apex:page javascript but the parameter is always null and I cannot figure out WHY?

I created a simple test page to test what I am doing:

--------------------------------------------------------------------------

<apex:page standardController="Quote" extensions="QuoteLineItems_ControllerExtension">

<apex:pagemessages />
<apex:pageBlock mode="edit">
    <apex:form id="testPage_Form">
        <script language="JavaScript" type="text/javascript">  
            function test1() {
                alert('>> init TEST_PAGE <<');   
                updateTest('this is my test data');
            }
        </script>
        
        <apex:actionFunction name="updateTest" action="{!updateTestData}">
           <apex:param name="test_param1" value="TEST_DUMMY" />
        </apex:actionFunction>
        
        <input type='button' value='TEST 1' onclick='test1();'/>
    </apex:form>
</apex:pageBlock>
</apex:page>

Here is a method in my the controller:

--------------------------------------------------

    public PageReference updateTestData() {
        System.Debug('>>> updateTest <<<');
        String test_param1 = ApexPages.CurrentPage().getParameters().get('test_param1');
        System.Debug('>>> test_param1 = '+ test_param1 + ' <<<');
        return null;
    }

Debug Log returns:

    >>> updateTest <<<

    >>> test_param1 = null <<<         ?? WHY NULL, expecting 'this is my test data'

WHAT am I doing wrong?