# Questions tagged [l-functions]

L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.

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### Explicit translation of Ramanujan’s condition for the Selberg Class?

According to many sites such as Wikipedia, the Ramanujan condition of the Selberg class can be stated as $$\boxed{\forall\epsilon>0:a_n\ll_\epsilon n^\epsilon}$$ My question is: what exactly does ...
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### q-series Expansion at Cusp and L-Functions

We know that the coefficients at $\infty$ of a modular form that is an eigenfunction of all Hecke operators in, say, $\Gamma_0(q)$, give an $L$-function with an Euler product and a functional equation....
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### Zeros of L function on the 0.5 line

Could someone please tell what results are known about the zeros of L function, $L(s,\chi)$ on the 1/2 line, where $\chi$ is a character mod $q$? Is there an upper bound for this count when we count ...
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### Does the relationship between the L function of the elliptic curve and a quadruple series hold?

Let $E_{X_0(11)}$ be the elliptic curve (over $\bf Q$) of conductor $11$ defined by $$y^2+y=x^3-x^2-10x-20.$$ First, some theorems and formulas are introduced as follows. The modularity theorem (Slow ...
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### Rankin-Selberg Convolution of newforms with different levels

Let $f \in \mathscr{S}_{k}(N,\chi)$ and $g \in \mathscr{S}_{k}(M,\psi)$ be newforms with $(N,M) = 1$. For $\Re(s) > 1$, I was able to derive an integral representation for the Rankin-Selberg ...
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