Dirichlet L-function ($L(s,\chi )$)

For $\mathrm{Re}s>1$,

For real $s>1$, Orthogonality of Dirichlet characters gives $\log {\mathcal{P}}_q(s)=\phi (q){\sum }_{p\nmid q}{\sum }_{j\ge 1,p^j\equiv 1(\mathrm{mod}q)}1/(j{p}^{js})\ge 0$. Thus ${\mathcal{P}}_q(s)\ge 1$. Only the principal Dirichlet character contributes a pole at one, and that pole is simple. A zero of a nonreal Dirichlet L-function at one brings a zero of its conjugate too; their two zeros overpower that pole and would make the product tend to zero. This proves nonvanishing for nonreal Dirichlet characters and is a useful complement to arguments for real Dirichlet characters.

For a primitive nonprincipal Dirichlet character of conductor $q$ and parity $a$, the completed Dirichlet L-function is entire of order one. Its genus-one Hadamard factorization gives the displayed logarithmic derivative, with $B_{\chi }$ the logarithmic derivative of the completion at zero and zeros counted with multiplicity. Combined summands converge locally normally away from zeros. For an imprimitive character induced by ${\chi }^{\ast }$, add ${\sum }_{p\mid Q,p\nmid q}{\chi }^{\ast }(p)\log p/(p^s-{\chi }^{\ast }(p))$ for the removed Euler factors. The primitive principal case is the global partial-fraction expansion of the zeta logarithmic derivative.

// Target: algebraic-number-theory.bigb

For a primitive nonprincipal Dirichlet character with parity $a$, the Gamma function poles in its completed Dirichlet L-function force zeros of $L(s,\chi )$ at $s=-a-2m$, $m=0,1,\dots$. The functional equation and nonvanishing of nonprincipal Dirichlet L-functions at one show that these zeros are simple and cancel the poles, rather than remaining zeros of the completion. For the Dirichlet beta function, the odd-parity examples are $\beta (-1)=\beta (-3)=\cdots =0$. The primitive principal conductor-one case is different: its zeta trivial zeros start at $-2$ and the completion has poles at zero and one.

The coefficients of $\zeta (s)L(s,\chi )$ are multiplicative. On $p^k$, their values are $k+1$ when $\chi (p)=1$, alternating one and zero when $\chi (p)=-1$, and one when $\chi (p)=0$. In particular $a_{m^2}\ge 1$. Their logarithmic Euler coefficients $\Lambda (n)(1+\chi (n))$ are also nonnegative, supporting uniqueness of a possible exceptional real Dirichlet zero.

For a nonprincipal Dirichlet character, the line is zero-free. Euler product positivity for L-function nonvanishing proves this for nonreal characters at every height, and for all characters at nonzero height, where a possibly principal squared-character factor has no pole. The point of height zero is the nonvanishing of nonprincipal Dirichlet L-functions at one. A principal L-function has a pole at one, and no zeros at the other points of this line.

The inducing primitive Dirichlet character has conductor of a Dirichlet character $q_0$. Increasing the modulus removes Euler factors only at new prime divisors. The correction creates zeros on the imaginary axis; raising the exponents of primes already dividing the conductor of a Dirichlet character changes no Euler factor. Primitive gamma factors continue to use $q_0$. For real-even characters this includes principal primitive conductor of a Dirichlet character one, with its Riemann zeta function pole treated separately.

For a primitive nonprincipal Dirichlet character, the theta Mellin integral makes this an entire function satisfying $\Lambda (s,\chi )={\epsilon }_{\chi }\Lambda (1-s,\overline{\chi })$. The gamma poles cancel the trivial zeros of a Dirichlet L-function. For the principal Dirichlet character whose conductor of a Dirichlet character is one the completion is meromorphic, with poles at zero and one.

For a primitive Dirichlet character whose conductor of a Dirichlet character is $q>1$ and whose character parity is $a$, Poisson summation gives ${\theta }_{\chi }(x)={\epsilon }_{\chi }{x}^{-a-1/2}{\theta }_{\overline{\chi }}(1/x)$, where ${\epsilon }_{\chi }=\tau (\chi )/(i^a\sqrt{q})$. Its rapid decay at both ends yields the entire Mellin representation of the completed Dirichlet L-function. The case of conductor of a Dirichlet character equal to one uses the ordinary Jacobi theta function and a subtracted constant term.

For a primitive Dirichlet character modulo $N>1$ of parity $a\in \{0,1\}$, put ${\Theta }_{\chi }(\tau )={\sum }_n{n}^a\chi (n)e^{\pi i{n}^2\tau }$, extending the character by zero on nonunits. The square root is analytic with positive value when $-i\tau$ is positive real. Apply Poisson summation separately in each residue class modulo $N$, and use the finite Fourier transform of a primitive Dirichlet character. The Fourier transforms of $e^{\pi i\tau x^2}$ and $x{e}^{\pi i\tau x^2}$ are $(-i\tau {)}^{-1/2}{e}^{-\pi i{t}^2/\tau }$ and $(t/\tau )(-i\tau {)}^{-1/2}{e}^{-\pi i{t}^2/\tau }$. These give the formula, including its odd-character phase.

Take the even primitive Dirichlet character ${\chi }_{12}$ whose values at $1,5,7,11$ modulo twelve are $1,-1,-1,1$. Its Gauss sum of a Dirichlet character is $\sqrt{12}$. The displayed series $A$ has $A(-1/\tau )=\sqrt{-i\tau }A(\tau )$ and $A(\tau +1)=e^{\pi i/12}A(\tau )$. Thus $A^{24}$ is the normalized weight-twelve modular discriminant. It has no interior zeros by the valence formula for the modular group. Its logarithmic derivative equals $E_2/24$, so integration gives the Dedekind eta function product. The odd character modulo four similarly produces a series $B$ with $B^8=A^{24}$ and leading term $q^{1/8}$; consequently $B=A^3$.

For a nonprincipal Dirichlet character modulo $N$, setThe numerator vanishes at $t=0$, so $H_{\chi }$ is analytic there and exponentially decreasing as $t\to \infty$. Initially for $\mathrm{Re}s>1$, $\Gamma (s)L(\chi ,s)={\int }_0^{\infty }{H}_{\chi }(t)t^{s-1}dt$. This integral already extends holomorphically to $\mathrm{Re}s>0$. Subtracting finite Taylor polynomials near zero, as in meromorphic continuation of a Mellin transform from an asymptotic expansion, continues it with possible simple poles only at nonpositive integers. Zeros of the reciprocal Gamma function cancel these, proving that $L(\chi ,s)$ is an entire function. This does not require a primitive Dirichlet character.