Zeta

Zeta (uppercase Ζ, lowercase ζ; Greek: Ζήτα [ziːta] Zita) is the sixth letter of the Greek alphabet. In the system of Greek numerals it has a value of 7. It was derived from the Phoenician letter Zayin . Letters that arose from Zeta include the Roman Z and Cyrillic З (Ze).

Unlike the other Greek letters, this letter did not take its name from the Phoenician letter it was derived from; it was given a new name on the pattern of beta, eta and theta.

Zeta has the numerical value 7 rather than 6 because the letter digamma (also called 'stigma' as a Greek numeral) was originally in the sixth position in the alphabet.

Zeta can be said to be the origin of the most common pronunciation of the Roman letter Z.

Symbol

The lower case letter can be used to represent:

The damping ratio of a spring-system in engineering and physics
The effective nuclear charge on an electron in quantum chemistry
The Riemann zeta function in mathematics
The lag angle in helicopter blade dynamics
Zoophilia

Pronunciation

The letter ζ represents a voiced alveolar fricative (IPA: [z]) in modern Greek.

The sound represented by zeta in Classical Greek is disputed. See Ancient Greek phonology and Pronunciation of Ancient Greek in teaching.

Most handbooks agree on attributing to it the pronunciation [zd] (like Mazda), but some scholars believe that it was an affricate [dz] (like Italian mezzo). The modern pronunciation was, in all likelihood, established in the Hellenistic age and was probably a common, if not exclusive, practice already in Classical Attic, considering that it could count as one or two consonants metrically in the Attic drama.

The arguments in favour of [zd]:

IE *zd becomes ζ in Greek (e.g. *sísdō > ἵζω). Contra: these words are rare and it is therefore more probable that *zd was absorbed by *dz (< *dj, *gj, *j).
Without [sd] there would be an empty space between [sb] and [sg] in the Greek sound system (πρεσβύς, σβέννυμι, φάσγανον), and a voiced affricate [dz] would not have a voiceless correspondent. Contra: a) words with [sb] and [sg] are rare; b) there was [sd] in ὅσδε, εἰσδέχται etc.; and c) there was in fact a voiceless correspondent in Archaic Greek ([ts] > Attic, Boeotian ττ, Ionic, Doric σσ).
Persian names with zd and z are transcribed with ζ and σ respectively in Classical Greek (e.g. Artavazda = Ἀρτάβαζος / Ἀρτάοζος ~ Zara(n)ka- = Σαράγγαι. Similarly, the Philistine city Ashdod was transcribed as Αζωτος.
ν disappears before ζ like before σ(σ), στ: e.g. *πλάνζω > πλᾰ́ζω, *σύνζυγος = *συνστέλλω > σῠστέλλω. Contra: ν may have disappeared before /dz/ if one accepts that it had the allophone [z] in that position like /ts/ had the allophone [s]: cf. Cretan ἴαττα ~ ἀποδίδονσα (Hinge).
Verbs beginning with ζ have ἐ- in the perfect reduplication like the verbs beginning with στ (e.g. ἔζηκα = ἔσταλται). Contra: a) The most prominent example of a verb beginning with στ has in fact ἑ- < *se- in the perfect reduplication (ἕστηκα; b) the words with /ts/ > σ(σ) also have ἐ-: Homer ἔσσυμαι, -ται, Ion. ἐσσημένῳ.
Alcman, Sappho, Alcaeus and Theocritus have σδ for Attic-Ionic ζ. Contra: The tradition would not have invented this special digraph for these poets if [zd] was the normal pronunciation in all Greek. Furthermore, this convention is not found in contemporary inscriptions, and the orthography of the manuscripts and papyri is Alexandrine rather than historical. Thus, (σδ indicates only a different pronunciation from Hellenistic Greek [z(ː)], i.e. either [zd] or [ʣ].
The grammarians Dionysius Thrax and Dionysius of Halicarnassus class ζ with the "double" (διπλᾶ) letters ψ, ξ and analyse it as σ + δ. Contra: The Roman grammarian Verrius Flaccus believed in the opposite sequence, δ + σ (in Velius Longus, De orthogr. 51), and Aristotle says that it was a matter of dispute (Metaph. 993a) (though Aristotle might as well be referring to a [zː] pronunciation).
Some Attic transcriptions of Asia Minor toponyms (βυζζαντειον, αζζειον, etc) show a -ζζ- for ζ; assuming that Attic value was [zd], it may be an attempt to transcribe a dialectal [dz] pronunciation; the reverse cannot be ruled completely, but a -σδ- transcription would have been more likely in this case. This suggest that different dialects had different pronunciations.
Some Attic inscriptions have -σζ- for -σδ- or -ζ-, which is thought to parallel -σστ- for -στ- and therefore to imply a [zd] pronunciation.

Arguments in favour of [dz] are:

The Greek inscriptions almost never write ζ in words like ὅσδε, τούσδε or εἰσδέχται, so there must have been a difference between this sound and the sound of ἵζω, Ἀθήναζε. Contra: a few inscriptions do seem to suggest that ζ was pronounced like σδ (though it may indeed be a minoritary pronunciation).
It seems improbable that Greek would invent a special symbol for the bisegmental combination [zd], which could be represented by σδ without any problems. /ds/, on the other hand, would have the same sequence of plosive and sibilant as the double letters of the Ionic alphabet ψ /ps/ and ξ /ks/, thereby avoiding a written plosive at the end of a syllable. Contra: the use of a special symbol for [zd] is no more or no less improbable that the use of ψ for [ps] and ξ for [ks], and such use of special letters may be justified by the fact that they are the only double sounds that could appear at a word initial.
Boeotian, Elean, Laconian and Cretan δδ are more easily explained as a direct development from *dz than through an intermediary *zd. Contra: a) the sound development dz > dd is improbable (Mendez Dosuna); b) ν has disappeared before ζ > δδ in Laconian πλαδδιῆν (Aristoph., Lys. 171, 990) and Boeotian σαλπίδδω (Sch. Lond. in Dion. Thrax 493), which suggests that these dialects have had a phase of metathesis (Teodorsson).
Greek in South Italy has preserved [dz] until modern times. Contra: a) this may be a later development from [zd] or [z]; b) even if it is derived from an ancient [dz], it may be a dialectal pronunciation.
Vulgar Latin inscriptions use the Greek letter Z for indigenous affricates (e.g. zeta = diaeta), and the Greek ζ is continued by a Romance affricate in the ending -ίζω > Italian. -eggiare, French -oyer. Contra: whether the prononciation of ζ was [dz], [zd] or [zː], di would probably still have been the closest native Latin sound.

In summary:

σδ is attested only in Lesbian and Spartan lyric poetry from the Archaic age and in Bucolic poetry from the Hellenistic Age. Most scholars would take this as an indication that the [zd]-pronunciation existed in the dialects of these authors.
The transcriptions from Persian by Xenophon and testimony by grammarians support the pronunciation [zd] in Classical Attic. On the other hand. The fact that ζ (e.g. ὄζω) and σδ (e.g. ὅσδε) are distinguished in all Classical inscriptions and literary texts indicates a different pronunciation.
[z(ː)] is attested from c. 350 BC in Attic inscriptions, and was the probable value in Koine.
[dʒ] or [dz] may have existed in some other dialects in parallel.

References

Allen, William Sidney: Vox Graeca: A guide to the pronunciation of Classical Greek. Cambridge University Press 1987, pp. 56-59.
Hinge, George: "Die Aussprache des griechischen Zeta", in: Die Sprache Alkmans: Textgeschichte und Sprachgeschichte (PhD dissertation), Aarhus 2001, pp. 212-234 = [1]
Méndez Dosuna, Julián: "On <Ζ> for <Δ> in Greek dialectal inscriptions", Die Sprache 35, 1993, pp. 82-114.
Rohlfs, Gerhard: "Die Aussprache des z (ζ) im Altgriechischen", Glotta 8, 1962, pp. 3-8.
Teodorsson, Sven-Tage: "The pronunciation of zeta in different Greek dialects". In: E. Crespo i.a. (ed.): Dialectologia Graeca: Actas del II Coloquio Internacional de Dialectología Griega. Universidad Autónoma de Madrid 1993, pp. 305-321.