The Wolfram MathWorld article Weisstein, Eric W. "Dirichlet L-Series." From MathWorld--A Wolfram Web Resource. mentions Hecke (1936) found a remarkable connection between each modular form with Fourier series
$$f(\tau )=c(0)+\sum_{n=1}^\infty c(n)\,e^{2 \pi i n \tau}\tag{1}$$
and the Dirichlet L-series
$$\phi(s)=\sum_{n=1}^\infty\frac{c(n)}{n^s}\tag{2}$$
This answer to a related question on Math StackExchange seems to contradict the Wolfram MathWorld article.
The Wolfram MathWorld article defines a Dirichlet L-series as a series of the form
$$L_k(s,\chi)=\sum_{n=1}^\infty\chi_k(n)\,n^{-s}\tag{3}$$
where $\chi_k(n)$ is a Dirichlet character whereas the answer on Math StackExchange claims "you will never get the coefficients of the Dirichlet series $\zeta(s)$ or $L(s, \chi)$ as the q-expansion coefficients of a modular form".
Note the Wolfram MathWorld article refers to $\phi(s)$ as a Dirichlet L-series which implies $c(n)$ is a Dirichlet character.
Question: What is the explanation for this seeming contradiction with respect to the relationship between modular forms and Dirichlet L-series?
For a Dirichlet character and $a = (1-\chi(-1))/2$ then $f(z)=\sum_n n^a \chi(n) e^{2i\pi nz}$ is a weight $a+1/2$ modular form (the functional equation of $L(s,\chi)$ follows from $f(z)=c z^{-a+1/2}f(-1/(qz))$). The weight $k/2$ forms are more complicated than those with integral weights, they don't deserve to be called genuine modular forms.
Any modular form decomposes as a finite sum of modular forms with multiplicative coefficients (eigenforms for the Hecke operators) whose corresponding Dirichlet series is (up to finitely Euler factors) in the Selberg class (ie. it keeps almost all the properties of $\zeta(s)$).
The modular forms are not only a way to create some new Selberg class L-functions, they also add many algebraic, analytic, geometrical, representation, Galois and modulo $p$ information to (some but not all) elements of the Selberg class. Making the correspondence precise leads to the Langlands program.
Obviously the Dirichlet L-functions are the simplest examples in the Selberg class, but on the modular side the simplest example becomes the Eisenstein series. The highly non-trivial example of modular form is $\Delta(z)=E_4(z)^3-E_6(z)^2$ which is important because it has only one simple zero at $i\infty$ so we can use it to factorize (cusp) forms of higher weight.
The must read text on modular forms is Diamond & Shurman.