I have already asked this question here in a different form, but really need an answer. Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc... (Selberg class if you like), of order 1, and let $\Lambda(s)$ be the completed $L$-function with gamma factors. We thus have $\Lambda(k-s)=\omega\Lambda^*(s)$, where $\Lambda^*$ is the "dual" Lambda function (example: if $L(s)$ corresponds to a Dirichlet character $\chi$, $\Lambda^*$ corresponds to its conjugate), and $\omega$ root number of modulus 1.

Assume for instance that there are no poles. Since $\Lambda$ has order $1$ it has a Hadamard product $$\Lambda(s)=ae^{bs}\prod_{\rho}(1-s/\rho)\;,$$ where the product is over the zeros of $\Lambda$ and understood as the limit as $T\to\infty$ of the product for $|\rho|<T$ (on purpose I do not use the more standard $(1-s/\rho)e^{s/\rho}$).

My question is this: do we always have $b=0$ ? This is trivial if $\Lambda^*=\Lambda$ (self-dual), otherwise the only thing I can prove is that $b$ is purely imaginary. I have experimented numerically with some non self-dual $L$ functions attached to Dirichlet characters, and it seems to be true.

Remarks: 1) I may have a proof using the "explicit formula" of Weil, but I am not sure of its validity, and it seems too complicated. 2) I have a vague memory of Harold Stark mentioning this result 50 years ago.