Fungrim entry: 8ff1ff

\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}(s)\right)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1

TeX:

\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}(s)\right)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("8ff1ff"),
    Formula(LessEqual(Abs(DirichletL(s, chi)), RiemannZeta(Re(s)))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 1))))

Topics using this entry

Dirichlet L-functions