what does $L(s, \pi \times \chi)$ mean in analytic number theory? Given a Dirichlet character -- a character $\chi: (\mathbb{Z}/p\mathbb{Z})^\times \to \mathbb{C}^\times$ one can define a Dirichlet L-function:
$$ L(s, \chi) = \sum \frac{\chi(n)}{n^s}$$
if $\pi$ is a representation of $\text{GL}(2, \mathbb{A})$ - a very complicated group based on the adèles, $\mathbb{A}$, one can also define an L-function, 
$$ L(s, \pi \times \chi) = \sum \dots $$
If I am not 100% sure of the definition of Adèle, there is discussion on Wikipedia roughy the product over all completions of $\mathbb{Z}$
$$ \mathbb{A}_\mathbb{Q} \simeq \Big( \hat{\mathbb{Z}} \times \mathbb{R} \Big)\otimes_\mathbb{Z} \mathbb{Q} $$
Presumably the general linear group is the set of $2\times 2$ invertible matrices with elements in this ring:
$$  \text{GL}(2, \mathbb{A}) \stackrel{?}{=} \left\{ \left( \begin{array}{cc} a & b \\ c & d\end{array}\right) : ad -bc = 1 \textbf{ and }a,b,c,d \in \mathbb{A}_\mathbb{Q}\right \} $$
I can even find textbooks on this standard material, although it is written in rather unforgiving abstract language.
$$ L(s, \pi \times \chi) = \sum \frac{(\pi \times \chi)(n)}{n^s} $$
This is my best guess except I can't think of any examples of $\pi$ and I thought $\pi$ was a function of $2 \times 2$ matrices in  $\text{GL}_2$ rather than integers in $\mathbb{Z}$.
 A: $\pi$ is an automorphic representation. Automorphic representations of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ correspond to modular forms. In particular, cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ are in a bijective correspondence with the union of the set of Maaß newforms of weight $0$ or $1$ and the set of holomorphic newforms of weight $k \geq 2$.
To read about modular forms, first look perhaps at the Wikipedia articles about them, then perhaps Iwaniec's books on automorphic forms. To see the link between modular forms and automorphic representations, see Goldfeld and Hundley or Bump's books.
Anyway, the key point is that associated to a Maaß newform $f$ or a holomorphic newform $f$ is a multiplicative function $\lambda_f(n)$; these are the eigenvalues of the Hecke operators acting on $f$, and are closely related to the coefficients of the Fourier expansion of $f$.
The $L$ function $L(s,\pi \times \chi)$ is then defined as
\[L(s,\pi \times \chi) = \sum_{n = 1}^{\infty} \frac{\lambda_f(n) \chi(n)}{n^s}.\]
(Actually, this is not quite true - associated to an automorphic representation or a modular form is a conductor $q \geq 1$, and if the conductor $q'$ of the Dirichlet character $\chi$ is not coprime to $q$, then this Dirichlet series needs to be slightly modified.)
In any case, you definitely should not be reading about automorphic representations without first having a very good grasp on modular forms.
