Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion

$$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$

and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke operators $T_p$, not just those with $(p,N) = 1$, and for the normalized L-function

$$L^{S_{\infty}}(f,s) = \sum\limits_{n=1}^{\infty} \frac{n^{(1-k)/2}a_n}{n^s}$$

one can add in an archimedean factor to make a completed L-function $L(f,s)$ which is an Euler product

$$L(f,s) = \prod\limits_{p \leq \infty} L_p(f,s)$$

and, for a suitable "contragredient" $g \in S_k(\Gamma_0(N))$ and epsilon root number $\epsilon(f,s)$, satisfies the functional equation

$$L(f,s) = \epsilon(f,s) L(g,1-s).$$

See for example Kowalski's article "Classical Automorphic Forms" in the book "An Introduction to the Langlands Program."

Now, $f$ can be identified in various ways with a cuspidal automorphic form on $\operatorname{GL}_2(\mathbb Q) \backslash \operatorname{GL}_2(\mathbb A_{\mathbb Q})$. There is a cuspidal automorphic representation $\pi = \otimes \pi_p$, unique up to isomorphism, which contains $f$. The normalized newform $f$ is conversely uniquely determined by $\pi$. This is one way to state the "Multiplicity One" theorem.

Also, $\pi$ itself has an L-function $L(\pi,s) = \prod\limits_{p \leq \infty} L(\pi_p,s)$ satisfying its own functional equation $L(\pi,s) = \epsilon(\pi,s) L(\tilde{\pi},1-s)$.

At the primes $p$ not dividing $N$, the representation $\pi_p$ is spherical, and the L-function $L(\pi_p,s)$ arises from the local spherical Hecke algebra of $\operatorname{GL}_2(\mathbb Q_p)$. Here things can be normalized so that

$$L(\pi_p,s) = L_p(f,s). \tag{1} $$
What about the primes dividing $N$? Here $L_p(f,s)$ arises from a Hecke operator $T_p$, but the representation $\pi_p$ could be supercuspidal, or it could have an Iwahori-fixed vector but be non-spherical. With the proper normalizations, can we have equation (1) holding for *all* primes $p$ (and consequently $L(\pi,s) = L(f,s)$)? Note that when $\pi_p$ is supercuspidal or is induced from a ramified character, $L(\pi_p,s) = 1$.

I expect it's possible to normalize things so that the local factors of $L(f,s)$ and $L(\pi,s)$ agree at all places, but I have never seen any reference which does this.