# primitive Dirichlet characters

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## 4 matching pages

##### 1: 27.8 Dirichlet Characters
27.8.6 $\sum_{r=1}^{\phi\left(k\right)}\chi_{r}\left(m\right)\overline{\chi}_{r}(n)=% \begin{cases}\phi\left(k\right),&m\equiv n\pmod{k},\\ 0,&\mbox{otherwise}.\end{cases}$
A Dirichlet character $\chi\pmod{k}$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor $d), there exists an integer $a\equiv 1\pmod{d}$, with $\left(a,k\right)=1$ and $\chi\left(a\right)\neq 1$. If $k$ is prime, then every nonprincipal character $\chi\pmod{k}$ is primitive. …
##### 2: 24.16 Generalizations
###### §24.16(ii) Character Analogs
Let $\chi$ be a primitive Dirichlet character $\mod f$ (see §27.8). …
##### 3: 27.10 Periodic Number-Theoretic Functions
The finite Fourier expansion of a primitive Dirichlet character $\chi\pmod{k}$ has the form …
##### 4: 25.15 Dirichlet $L$-functions
When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: … Since $L\left(s,\chi\right)\neq 0$ if $\Re s>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $L\left(s,\chi\right)$ for $\Re s<0$ (the so-called trivial zeros) are as follows: …