Modular first Chebyshev function

Definition

Let $x$ be a positive real number, $n$ be a natural number, and $a$ be an integer. The modular first Chebyshev function of $x$, denoted $\vartheta (x;n,a)$ or $\theta (x;n,a)$, is defined as:

$\vartheta (x;n,a)=\sum _{p\leq x,p\equiv a{\pmod {n}}}\log p$,

where the sum is only over the prime numbers less than or equal to $x$.

The is the modular version of the first Chebyshev function.