NotationsFor $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written in the form \begin{equation} f(s)=\sum_{n=1}^{\infty}{\frac{a_n}{n^s}} \nonumber \end{equation} where the series is convergent when $\operatorname{Re}(s)$ is large enough.

I recently found the following result on $L$-functions :

**Theorem**
Let $\gamma$ be a meromorphic function on $\textbf{C}$, $k$ a real. Suppose that $L_1$ and $L_2$ are two meromorphic functions on $\textbf{C}$ such that $L_i$ (for $i=1$ and $2$) satisfies the following conditions

$(i)$ There exists a meromorphic function $L_i^*$ on $\textbf{C}$ and a positive real $N_i$ such that \begin{equation} L_i(k-s)=N_i^s\gamma(s)L_i^*(s) \quad (\forall s\in\textbf{C}), \nonumber \end{equation} $(ii)$ there exists a polynomial $P_i$ such that $P_iL_i$ is holomorphic and of finite order in $\textbf{C}$.

Assume further that $L_1/L_2$ has only finitely many poles and that $L_1/L_2$ and $L_1^*/L_2^*$ are represented by ODS.

Then there exist a positive integer $N$ and complex numbers $(a_d)_{d|N}$ such that \begin{equation} L_1(s)=\left(\sum_{u|N}{\frac{a_u}{u^s}}\right)L_2(s) \quad (\forall s\in \textbf{C}). \nonumber \end{equation}

I have also found interesting applications (certainly already known) :

**Corollary $1$** Let $\chi$ and $\varphi$ be two distinct primitive Dirichlet characters. Then $L(\chi,s)/L(\varphi,s)$ has infinitely many poles.

**Corollary $2$** Let $f$ and $g$ be two linearly independent Hecke forms of integer weight $k$ with respect to $\Gamma_0(N)$. Then $L(f,s)/L(g,s)$ has infinitely many poles.

I need an answer to those few questions :

$1.$ Is this theorem known ?

$2.$ If yes, does it deserve a publication ?

$3.$ Are there interesting references on that subject ?

I should add that I am a french graduate student (I apologize for my approximative english...) and I worked on number theory on my own : I hope that my problem is not too badly written. Many thanks !

directlymention $L_1$ and $L_2$. It would be better to write those conditions using "$L_i$ for $i = 1$ and $2$" in each and writing $L_i$ instead of $L$ in the equations. $\endgroup$