We study the Hardy space of translated Dirichlet series $${\mathscr {H}}_{+}$$ . It consists on those $$\sum a_n n^{-s}$$ such that for some (equivalently, every) $$1 \le p < \infty $$ , translation {a_{n}}n^{-(s+\frac{1}{\sigma })}$$ belongs to {H}}^{p}$$ every $$\sigma >0$$ prove this set, endowed with topology induced by seminorms $$\left\{ \Vert \cdot _{2,k}\right\} _{k\in {\mathbb {N}}}$$ ...