Let $\zeta _K(s)$ denote the Dedekind zeta-function associated to a number field K. We give an effective upper bound for the height of the first nontrivial zero other than $1/2$ of $\zeta _K(s)$ under the generalised Riemann hypothesis. This is a refinement of the earlier bound obtained by Sami [‘Majoration du premier zéro de la fonction zêta de Dedekind’, Acta Arith. 99(1) (2000), 61–65].

Let $F_{2^n}$ be the Frobenius group of degree $2^n$ and of order $2^n ( 2^n-1)$ with $n \ge 4$ . We show that if $K/\mathbb {Q} $ is a Galois extension whose Galois group is isomorphic to $F_{2^n}$ , then there are $\dfrac {2^{n-1} +(-1)^n }{3}$ intermediate fields of $K/\mathbb {Q} $ of degree $4 (2^n-1)$ such that they are not conjugate over $\mathbb {Q}$ but arithmetically equivalent over $\mathbb {Q}$ . We also give an explicit method to construct these arithmetically equivalent fields.

We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

We give upper bounds on the modulus of the values at $s\,=\,1$ of Artin $L$-functions of abelian extensions unramified at all the infinite places. We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for $\text{CM}$-fields. For example, we will reduce the determination of all the non-abelian normal $\text{CM}$-fields of degree 24 with Galois group $\text{S}{{\text{L}}_{\text{2}}}\left( {{F}_{3}} \right)$ (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of 23 such $\text{CM}$-fields.