Shimizu L-function

<h2 id="definition">Definition</h2>
Suppose that K is a totally real algebraic number field, M is a lattice in the field, and V is a subgroup of maximal rank of the group of totally positive units preserving the lattice. The Shimizu L-series is given by

L
        (
        M
        ,
        V
        ,
        s
        )
        =
        
          ∑
          
            μ
            ∈
            {
            M
            −
            0
            }
            
              /
            
            V
          
        
        
          
            
              sign
              ⁡
              N
              (
              μ
              )
            
            
              
                |
              
              N
              (
              μ
              )
              
                
                  |
                
                
                  s
                
              
            
          
        
      
    
    {\displaystyle L(M,V,s)=\sum _{\mu \in \{M-0\}/V}{\frac {\operatorname {sign} N(\mu )}{|N(\mu )|^{s}}}}

<ul><li><a href="/facts/Michael_Atiyah/RQVJ9ATE">Atiyah, Michael Francis</a>; Donnelly, H.; Singer, I. M. (1982), "Geometry and analysis of Shimizu L-functions", <a href="/facts/Proceedings_of_the_National_Academy_of_Sciences/w75CThX0">Proceedings of the National Academy of Sciences of the United States of America</a>, 79 (18): 5751, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1982PNAS...79.5751A">1982PNAS...79.5751A</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1073%2Fpnas.79.18.5751">10.1073/pnas.79.18.5751</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/12685">12685</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0674920">0674920</a>, <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC346984">346984</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/16593231">16593231</a></li>
<li><a href="/facts/Michael_Atiyah/RQVJ9ATE">Atiyah, Michael Francis</a>; Donnelly, H.; Singer, I. M. (1983), "Eta invariants, signature defects of cusps, and values of L-functions", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, Second Series, 118 (1): 131–177, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2006957">10.2307/2006957</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2006957">2006957</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0707164">0707164</a></li>
<li>Shimizu, Hideo (1963), "On discontinuous groups operating on the product of the upper half planes", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, Second Series, 77 (1): 33–71, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970201">10.2307/1970201</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970201">1970201</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0145106">0145106</a></li></ul>

Shimizu L-function open-in-new

Shimizu L-function