I found that the normalisation of $L$-functions of classical modular forms and corresponding automorphic representations is somewhat confusing for me. Recall that if $f\in S_k(\Gamma_0(N))$ is a classical cusp form with respect to $\Gamma_0(N)$, then $f$ has a $q$-expansion $f=\sum a_n q^n$, and the (classical) partial $L$-function (non-archimedean part) of $f$ is defined as $$L(s,f)=\sum \frac{a_n}{n^{s}}.$$ After adding an archimedean $L$-factor, we obtain the completed $L$-function $\Lambda(s,f)$. If $f$ is a newform (Hecke eigenform and orthogonal to oldforms), then we know that $\Lambda(s,f)$satisfies a functional equation, with central line $Re(s)=k/2$, c.f. GTM228, Diamond-Shurman section 5.10.
On the other hand, we can lift $f$ as an automorphic form on $G(\mathbb{Q})\backslash G(\mathbb{A})$, where $\mathbb{A}=\mathbb{A}_\mathbb{Q}$, $G:=GL(2)$. This process is rather standard: we have to start from the automorphy factor $$j(g, \tau)=\operatorname{det}(g)^{-\frac{1}{2}}(c \tau+d), \quad g=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) ,$$ and lift it to $GL(2,\mathbb{R})^+$ by the following formula $$\phi\left(g_{\infty}\right)=\phi_f\left(g_{\infty}\right)=f\left(g_{\infty}(i)\right) j\left(g_{\infty}, i\right)^{-k},$$ and finally we use the strong approximation to obtain $$G(\mathbb{Q}) \backslash G(\mathbb{A}) / K^{\prime} \simeq \Gamma^{\prime} \backslash G(\mathbb{R})^{+},$$ where $$ K^{\prime}=K_0(N)=\left\{k=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid c \equiv 0 \bmod (N)\right\} $$ and $ \Gamma^{\prime}=G(\mathbb{Q}) \cap G(\mathbb{R})^{+} K^{\prime}=\Gamma_0(N) $. Then we obtain an adelic automorphic form $\phi$ and it generates an automorphic representation. Finally one defines its **automorphic $L$-function **according to the Godement-Jacquet theory of zeta integrals. Here the zeta interal is defined as $$ Z(\phi, h, s)=\int_{\mathrm{GL}_2(\mathbb{A})} \phi(g) h(g)|\operatorname{det}(g)|^{s+1 / 2} \mathrm{~d} g, $$ where $h(g)$ is a matrix coefficient of this automorphic representation. In some sense the $L$-function $L(s,\pi_f)$ is the "greatest common denominator" of these integrals (but here there's a huge bunch of tough technical problems, which I didn't mention here).
What makes me puzzled is: finally we have $$L(s,\pi_f)=\Lambda(s+(k-1)/2,f).$$ Here the left hand side is independent of the **weight ** of the modular form and the right hand side does depend on the weight of $f$. Intuitively, the weight is "packaged" in the lifting formula given above, and finally the weight $k$ appeared in the action of $K_\infty = O(2)\subset G(\mathbb{R})$, namely the type of the archimedean $(\mathfrak{g}_\infty,K_\infty)$-module. If the modern adelic theory generalises the classical theory, then I feel confused that left hand side is independent of the weight $k$. Because the normalisation of the automorphic $L$-function only requires a translation $1/2$ in the variable $s$ which has nothing to do with $k$, but the classical $L$-function is not the case!
Can any veteran explain this to me in some detail? I found the notes by Buzzard before: https://www.ma.imperial.ac.uk/~buzzard/maths/research/notes/automorphic_forms_for_gl2_over_Q.pdf but it seems not to explain my question here: the central character comes from the "nebentype" character of the modular form, not the weight?
Thanks a lot in advance!
