Let $L(s,\chi_i)$ be some Dirichlet $L$-functions. Let

$$F(s)=\int_{\Re w=\sigma} \frac{1}{w} L(w,\chi_1) L(s-w,\chi_2) dw$$

for some $\sigma>1$. When $\Re s-\sigma>1$, one can write the $L$-function as the corresponding $L$-series, and one can see from there that $F(s)$ is well-defined for $\Re s>2$.

Question: Do we have an analytic continuation of $F(s)$ beyond $\Re s=2$? If so, how to write it explicitly?

When $\chi_1=\chi_2$, one can simply do the change of variable $w\mapsto s-w$, but I don't see a way that works in general.

Let $L_i(s)$ be some $L$-functions. (I am interested in the case when $L_1, L_2$ are two different Hecke $L$-function associated to the same number field.) Let

$$F(s)=\int_{\Re w=\sigma} \frac{1}{w} L_1(w) L_2(s-w) dw$$

for some $\sigma>1$. When $\Re s-\sigma>1$, one can write the $L$-function as the corresponding $L$-series, and one can see from there that $F(s)$ is well-defined for $\Re s>2$.

Question: Do we have an analytic continuation of $F(s)$ beyond $\Re s=2$? If so, how to write it explicitly? Add conditions to the $L$-function if needed.

Edit: Some comment pointed out that if both $L$-functions are Dirichlet $L$-function, i.e. $L_i(s)=L(s,\chi_i)$, then there is a way to write it out.

When $L_1=L_2$, one can simply do the change of variable $w\mapsto s-w$, but I don't see a way that works in general.