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In a recent work arXiv:2004.14450, it has been shown that $L$-functions associated with arbitrary non-zero cusp forms take large values at the central critical point. The goal of this note is to derive analogous results for twists of Dirichlet-type functions. More precisely, for an odd integer $q >1$, let $F$ be a non-zero $\mathbb{C}$-linear combination of primitive, complex, even Dirichlet characters of conductor $q$. We show that for any $\epsilon>0$ and sufficiently large $X$, there are $\gg X^{1-\epsilon}$ fundamental discriminants $8d$ with $X < d \leq 2X$ and ${(d, 2q)=1}$ such that ${|L(1/2, F \otimes \chi_{8d})| }$ is large.

In 2008, Soundararajan showed that there exists a normalized Hecke eigenform $f$ of weight $k$ and level one such that $$ L(1/2, f ) ~\geq~ \exp\Bigg( (1 + o(1)) \sqrt{\frac{2\log k}{\log\log k} }\Bigg) $$ for sufficiently large $k \equiv 0 \pmod{4}$. In this note, we show that for any $\epsilon>0$ and for all sufficiently large $k \equiv 0 \pmod{4}$, the number of normalized Hecke eigenforms of weight $k$ and level one for which $$ L(1/2, f ) ~\geq~ \exp\left(1.41\sqrt{ \frac{ \log k }{\log\log k} }\right) $$ is $\gg_{\epsilon} k^{1-\epsilon}$. For an odd fundamental discriminant $D$, let $B_{k}(|D|)$ be the set of all cuspidal normalized Hecke eigenforms of weight $k$ and level dividing $|D|$. When the real primitive Dirichlet character $\chi_D$ satisfies $\chi_D(-1)= i^k$, we investigate the number of $f \in B_{k}(|D|)$ for which $L(1/2, f \otimes \chi_D)$ takes extremal values.

In 2011, Gun, Murty and Rath studied non-vanishing and transcendental nature of special values of a varying class of $L$-functions and their derivatives. This led to a number of works by several authors in different set-ups including studying higher derivatives. However, all these works were focused around the central point of the critical strip. In this article, we extend the study to arbitrary points in the critical strip.

We introduce some generalizations of the Euler-Kronecker constant of a number field and study their arithmetic nature., Comment: 8 pages

For a number field $K$, the Euler-Kronecker constant $\gamma_K$ associated to $K$ is an arithmetic invariant the size and nature of which is linked to some of the deepest questions in number theory. This theme was given impetus by Ihara who obtained bounds, both unconditional as well as under GRH for Dedekind zeta functions. In this note, we study the analogous constants associated to the narrow ray class fields of an imaginary quadratic field. Our goal for studying such families is twofold. First to show that for such families, the conditional bounds obtained by Ihara can be improved on the average, again under GRH for Dedekind zeta functions. Further, our family of number fields are non-abelian while such average bounds have earlier been studied for cyclotomic fields. The technical part of our work is to make the dependence of these upper bounds on the ambient number field explicit. Such explicit dependence is essential to further our objective., Comment: 36 pages