... τ)=k∈Zeπi(k+α) 2 τ +2 i(k+α)(w+β), and setφαβ(w, τ)=eπw 2 /2Im(τ)ϑαβ(w, τ).Annals of Mathematics Divisibility of anticyclotomic L -functions and theta functions with complex ... DIVISIBILITY OF ANTICYCLOTOMIC L -FUNCTIONS 785Lemma 3 .2. Let L, H and ψ be as above,(ω1,ω 2 ) a basis of L such thatIm(τ) > 0 for τ = ω 2 /ω1, and α0 and β0real numbers withψ(aω1+ bω 2 )=eπin(ab+2aα0+2bβ0).Then ... situation of Lemma3 .2 (Aaω1+bω 2 gj)(0) = eπi((α0+j/n−1 /2) (n(b−β0)+1 /2) +a(nβ0+1 /2) +1 /2) ×η(nτ)ga+α0+j/n−1 /2, n(b−β0)−1 /2 (nτ).Now the first assertion follows, since the Siegel functions...