Fungrim entry: 9b3fde

\left|L\!\left(s, \chi\right)\right| \le {\left(\frac{q \left|1 + s\right|}{2 \pi}\right)}^{\left( 1 + \eta - \operatorname{Re}(s) \right) / 2} \zeta\!\left(1 + \eta\right)

Assumptions:

q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \left(0, \frac{1}{2}\right] \;\mathbin{\operatorname{and}}\; -\eta \le \operatorname{Re}(s) \le 1 + \eta

References:

H. Rademacher, On the Phragmén-Lindelöf theorem and some applications, Mathematische Zeitschrift, December 1959, Volume 72, Issue 1, pp 192-204. Theorem 3. https://doi.org/10.1007/BF01162949

TeX:

\left|L\!\left(s, \chi\right)\right| \le {\left(\frac{q \left|1 + s\right|}{2 \pi}\right)}^{\left( 1 + \eta - \operatorname{Re}(s) \right) / 2} \zeta\!\left(1 + \eta\right)

q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \left(0, \frac{1}{2}\right] \;\mathbin{\operatorname{and}}\; -\eta \le \operatorname{Re}(s) \le 1 + \eta

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Re`	$\operatorname{Re}(z)$	Real part
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval

Source code for this entry:

Entry(ID("9b3fde"),
    Formula(LessEqual(Abs(DirichletL(s, chi)), Mul(Pow(Div(Mul(q, Abs(Add(1, s))), Mul(2, Pi)), Div(Sub(Add(1, eta), Re(s)), 2)), RiemannZeta(Add(1, eta))))),
    Variables(q, chi, s, eta),
    Assumptions(And(Element(q, ZZGreaterEqual(2)), Element(chi, PrimitiveDirichletCharacters(q)), Element(s, CC), Element(eta, OpenClosedInterval(0, Div(1, 2))), LessEqual(Neg(eta), Re(s), Add(1, eta)))),
    References("H. Rademacher, On the Phragmén-Lindelöf theorem and some applications, Mathematische Zeitschrift, December 1959, Volume 72, Issue 1, pp 192-204. Theorem 3. https://doi.org/10.1007/BF01162949"))

Topics using this entry

Dirichlet L-functions