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Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and $\lambda_1,\lambda_2$ some positive integers.

I would like to ask if there is an entire function $f$ such that $$ f(z+ v_1) = f(z) e^{2\pi i \lambda_1 Cz/3 }$$ and $$ f(z+ v_1) = f(z) e^{2\pi i \lambda_2 Cz/3 }$$ where $C \in \mathbb R$ is a constant such that $\frac{\lambda_1 \lambda_2 C (\zeta_2 -\zeta_1)}{3}=1$. This looks a bit like periodic boundary conditions, but since $z \in \mathbb C$ the modulus of these boundary conditions has of course a growing/decaying direction.

This looks pretty similar to something related to theta functions, but I don't quite get it together, as the underlying lattice looks rather different.

Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and $\lambda_1,\lambda_2$ some positive integers.

I would like to ask if there is an entire function $f$ such that $$ f(z+ v_1) = f(z) e^{2\pi i \lambda_1 Cz/3 }$$ and $$ f(z+ v_2) = f(z) e^{2\pi i \lambda_2 Cz/3 }$$ where $C \in \mathbb R$ is a constant such that $\frac{\lambda_1 \lambda_2 C (\zeta_2 -\zeta_1)}{3}=1$. This looks a bit like periodic boundary conditions, but since $z \in \mathbb C$ the modulus of these boundary conditions has of course a growing/decaying direction.

This looks pretty similar to something related to theta functions, but I don't quite get it together, as the underlying lattice looks rather different.