Questions tagged [l-functions]

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

10 questions
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Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$. $$\begin{array}{c|ccr} & 0 & 1 & 2 \\ \hline \chi_1 & 0 & 1 & 1 \\ \chi_2 & 0 & 1 & -1 \end{... 1answer 92 views Rankin-Selberg convolution : normalization issues I was told several times on MO that if  F : s\mapsto\sum_{n>0}\frac{a_{n}}{n^{s}} and  G : s\mapsto\sum_{n>0}\frac{b_{n}}{n^{s}}  for  \Re(s)>1  are L-functions, then provided the ... 1answer 2k views what does the “L” in “L-function” stand for? I haven't been able to find a reference that tells what word (if a word) the L is short for. 1answer 445 views L-function of an elliptic curve and isomorphism class Let E be an elliptic curve defined over \mathbb{Q}. We have a L-function$$L(E,s) built from the local parameters $a_p(E)$. If two elliptic curves are isomorphic, they clearly have the same $... 1answer 606 views $L$-functions of elliptic curves over$\mathbb{Q}$How to find out the$P_{v}(E/\mathbb{Q},X)$theoretically given below in the definition of$L$-functions for elliptic curves over$\mathbb{Q}?$Please cite some references for the same. For an ... 1answer 89 views Why does the determinant come in for Artin L-Functions? Let$L/K$be a Galois extension of number fields. For$\mathfrak p$a prime of$K$, unramified in$L$, the Frobenius elements$\sigma_{\mathfrak P}$for$\mathfrak P \mid \mathfrak p$are conjugate, ... 1answer 144 views how to construct the Z-function corresponding to Davenport-Helbron L-function? See https://aimath.org/news/gl3/zfunction.html Actually, this question doesnt make sense, the Davenport-Heilbronn "zeta function" isn't even an L-series so it isn't right to call it a "zeta function" ... 0answers 81 views definition of the L-function$L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is$f\$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...