# Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?

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.

• You know the exact functional equation satisfied by $L(f,s)$ from the Fricke involution, you want $L(s,\pi)$ to satisfy a similar one, which implies they are equal (since changing finitely primes adds some Euler factors in the functional equation), morally being left invariant under $GL_2(\Bbb{Q})$ is really the functional equation. – reuns Dec 26 '19 at 16:00

$$L(\pi,s)$$ agrees with $$L(f,s)$$ if $$f\in\pi$$ is a newform, and this is even true for $$\mathrm{GL}_n$$. Of course, things are complicated by the fact that there are many ways to define $$L(\pi,s)$$ and $$L(f,s)$$. I found the following papers very useful to check various consistencies: Schmidt and Kondo-Yasuda.
Here is a summary based on Kondo-Yasuda. Let $$G:=\mathrm{GL}_n$$. Let $$(\pi,V_\pi)$$ be a cuspidal automorphic representation of $$G$$ over $$\mathbb{Q}$$ of unitary central character $$\omega$$. Let $$K_p(c)$$ denote the subgroup of elements of $$G(\mathbb{Z}_p)$$ whose bottom row is congruent to $$\begin{pmatrix}0 & \cdots & 0 & 1\end{pmatrix}$$ modulo $$c$$. Let $$K(c):=\prod_p K_p(c)$$. The conductor $$c_\pi$$ is the smallest $$c$$ such that $$V_\pi^{K(c)}\neq\{0\}$$. An element of $$V_\pi^{K(c_\pi)}$$ is called a global newform: it is an eigenfunction of the convolution by any element of $$C_c(K_p(c_\pi)\backslash G(\mathbb{Q}_p)/K_p(c_\pi))$$. Let $$c_\omega$$ be the conductor of $$\omega$$, and consider the corresponding primitive Dirichlet character modulo $$c_\omega$$: $$\chi_\omega(m):=\omega(1,\underbrace{1,\dotsc,1}_{v\mid c_\omega},\underbrace{m,m,\dotsc}_{v\nmid c_\omega})= \omega(1,\underbrace{m^{-1},\dotsc,m^{-1}}_{v\mid c_\omega},\underbrace{1,1,\dotsc}_{v\nmid c_\omega}),$$ for $$m>0$$ and $$(m,c_\omega)=1$$. Note that $$c_\omega\mid c_\pi$$. Let $$\chi_\pi$$ denote the Dirichlet character modulo $$c_\pi$$ induced by $$\chi_\omega$$. For $$k\in\{1,\dotsc,n-1\}$$, we define the $$k$$-th Hecke operator at $$p$$ as the characteristic function of $$H_k(p):=K_p(c_\pi)\,\mathrm{diag}(\underbrace{p,\dotsc,p}_\text{k entries},\underbrace{1,\dotsc, 1}_\text{n-k entries})\,K_p(c_\pi).$$ Accordingly, let $$\lambda_{\pi,k}(p)$$ be the $$k$$-th normalized Hecke eigenvalue at $$p$$: $$\int_{H_k(p)}f(xg)\,dg=\lambda_{\pi,k}(p)p^{\frac{k(n-k)}{2}}\mathrm{vol}(K_p)f(x),\qquad f\in V_\pi^{K(c_\pi)}.$$ Further, let $$\lambda_{\pi,0}(p):=1$$ and $$\lambda_{\pi,n}(p):=\chi_\pi(p)$$.
Theorem (Tamagawa 1963, Satake 1963, Kondo-Yasuda 2010). $$L(\pi,s)=\prod_p\left(\sum_{k=0}^n(-1)^k\lambda_{\pi,k}(p)p^{-ks}\right)^{-1}.$$
• Thanks very much for your answer. I assume the archimedean equality $L_{\infty}(f,s) = L(\pi_{\infty},s)$ is easy to check ($\pi_{\infty}$ for a newform should be like a discrete series representation) – D_S Dec 25 '19 at 21:49
• @D_S: Yes. As a rule of thumb, as long as the $L$-function "looks good", i.e., it has an Euler product decomposition and a functional equation of the usual shape, "it is good". If the definition is not OK, it will show up as a problem in the analytic properties. – GH from MO Dec 25 '19 at 22:44
• For $L(\pi_\infty,s)$ I recommend Tate's article "Number theoretic background" in Volume 2 of Borel-Casselman (eds.): Automorphic forms, representations, and $L$-functions. – GH from MO Dec 25 '19 at 22:47