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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.

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  • $\begingroup$ using that $\langle f ,g \rangle = \langle \hat{f},\hat{g} \rangle$ I get something like $\sum_{n=1}^\infty \chi_2(n)n^{-s} (\sum_{k < n} \chi_1(k)) $. Since $a(n) = \chi_2(n)\sum_{k < n} \chi_1(k)$ is periodic, $\sum_{n=1}^\infty a(n) n^{-s}$ is entire except a simple pole at $s=1$ of residue the mean value of $a(n)$ $\endgroup$
    – reuns
    Commented Oct 12, 2016 at 23:40
  • $\begingroup$ @user1952009 Thanks for the comment. I am actually looking at the case when the L-functions are Hecke L-functions. Edited the question. $\endgroup$
    – Ted Mao
    Commented Oct 13, 2016 at 0:02

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