Fungrim entry: bc755b

\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} L\!\left(s, \chi\right) = \left(\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}(s) \le 0} L\!\left(s, \chi\right)\right) \cup \left\{ \rho_{n,\chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} L\!\left(s, \chi\right) = \left(\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}(s) \le 0} L\!\left(s, \chi\right)\right) \cup \left\{ \rho_{n,\chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`DirichletLZero`	$\rho_{n,\chi}$	Nontrivial zero of Dirichlet L-function
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("bc755b"),
    Formula(Equal(Zeros(DirichletL(s, chi), ForElement(s, CC)), Union(Parentheses(Zeros(DirichletL(s, chi), For(s), And(Element(s, CC), LessEqual(Re(s), 0)))), Set(DirichletLZero(n, chi), For(n), Element(n, SetMinus(ZZ, Set(0))))))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

Topics using this entry

Dirichlet L-functions