A specialisation of the Bump-Friedberg $L$-function

We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+\alpha ,2s),s\in \text{C}\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+\alpha ,\pi )L(2s,{\mathrm{\Lambda }}^2,\pi )$ which we denote by $L^{lin}(s,\pi ,\alpha )$ for this abstract, when $\pi$ is a cuspidal automorphic representation of $GL(k,A)$ for $A$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $\pi$, the partial $L$-function $L^{lin,S}(s,\pi ,\alpha )$ has a pole at 1/2, if and only if $\pi$ admits a (twisted) global period, this gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that $\pi$ has a twisted global period if and only if $L(\alpha +1\mathrm{/}2,\pi )\mathrm{\ne }0$ and $L(1,{\mathrm{\Lambda }}^2,\pi )=\mathrm{\infty }$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\geq 1/2$when$Re(\alpha)$is$\geq 0$.