Dirichlet characters and L-functions

A Dirichlet character modulo a positive integer $q$ starts as a group homomorphism from $(\mathbb Z/q\mathbb Z)^\times$ to the nonzero complex numbers. Extend it periodically to the integers and assign zero whenever the argument is not coprime to the modulus. The nonzero values have modulus one and are roots of unity. This extension is completely multiplicative, including products involving nonunits, because a product is a unit exactly when both factors are units.

The principal character is one on units and zero elsewhere; it is not the constant-one arithmetic function unless the modulus is one. For a nonprincipal character, summing over a complete residue period gives zero. Indeed, multiplication by a unit with character value different from one permutes the unit classes but multiplies their sum by that value. The partial sums are consequently bounded in absolute value by the modulus. Their cancellation will matter analytically even though the absolute values of the nonzero coefficients never decay.

A character modulo $q$ is induced from a character modulo a divisor when their values agree on all integers coprime to $q$. The unique minimal such modulus is its conductor, and the inducing character at that modulus is primitive. Extending the modulus can turn some formerly nonzero values into zeros, because additional primes become nonunits. The L-function therefore changes by removing precisely those Euler factors. Primes already dividing the conductor need no additional factor: their primitive character values were already zero.

The finite product in the displayed identity is first justified to the right of one, then continues wherever the primitive L-function does. Its factors contain the primitive character values; replacing all those values by one is generally wrong. Functional equations with conductor-dependent gamma factors are naturally stated for primitive characters. An imprimitive L-function inherits the primitive equation together with these extra finite factors, so blindly substituting its larger modulus into a primitive formula can create incorrect constants and zeros.

Partial summation with bounded character partial sums shows that a nonprincipal character series converges when the real part is positive, locally uniformly away from that boundary. Absolute convergence still requires real part greater than one: on units its absolute coefficients equal one. The primitive functional equation continues a nonprincipal L-function to an entire function, and the finite-factor identity gives the same entireness for an imprimitive nonprincipal character. Conditional convergence of the original series and analytic continuation beyond it remain distinct statements.

For the principal character, $L(s,\chi_0)=\zeta(s)\prod_{p\mid q}(1-p^{-s})$, with a simple pole at one of residue $\varphi(q)/q$. A primitive nonprincipal character has parity $a\in\{0,1\}$ determined by $\chi(-1)=(-1)^a$; its completed expression uses $(f/\pi)^{(s+a)/2}\Gamma((s+a)/2)L(s,\chi)$. Its reflection relates the character to its complex conjugate, rather than always to itself. In particular, a complex character need not have its own zeros symmetric across the real axis; conjugation can transfer them to another character’s L-function.