I research the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For Owen Murtagh B 1826 +/- 2 yrs, Co. Louth, with the father forenamed Mathew, I found 2 rcds: 2-1826 both in Darver parish. These are likely duplicate rcds.
For John Murtagh B 1834 ditto, with the father's forename unknown, I found 1 rcd in 1834. Father's forename was John.
For Denis Murtagh B 1825 ditto, I found 0 rcds.
For Mat(hew) Murtagh M 1824 +/- 1 yr in Co. Louth, I found 0 rcds.
For Mat(hew) Murtagh B 1800 +/- 4 yrs somewhere in Ireland with the father's forename unknown, I found 3 rcds: 1797 Westmeath and 2-1800 Louth. The father's forenames were Michael, Patrick, and Eugene respectively. Parishes were Delvin, Darver, and Darver.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.