Properties

Degree 1
Conductor $ 5 \cdot 11 $
Sign $0.486 - 0.873i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.587 − 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (0.587 + 0.809i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (−0.951 − 0.309i)18-s + (−0.309 − 0.951i)19-s + 21-s + ⋯
L(s,χ)  = 1  + (−0.587 − 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (0.587 + 0.809i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (−0.951 − 0.309i)18-s + (−0.309 − 0.951i)19-s + 21-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.486 - 0.873i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.486 - 0.873i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(55\)    =    \(5 \cdot 11\)
\( \varepsilon \)  =  $0.486 - 0.873i$
motivic weight  =  \(0\)
character  :  $\chi_{55} (27, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 55,\ (1:\ ),\ 0.486 - 0.873i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.492593171 - 0.8775273885i$
$L(\frac12,\chi)$  $\approx$  $1.492593171 - 0.8775273885i$
$L(\chi,1)$  $\approx$  1.134410250 - 0.4760026104i
$L(1,\chi)$  $\approx$  1.134410250 - 0.4760026104i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−33.102647145561186220004923671876, −32.17939025580797655974000544206, −31.01410630007374458687430639255, −29.79700067056472024929188400146, −27.882745045618870381991344122353, −27.33459567641497503809013521224, −26.14904011330336230870423152494, −25.266962258399316822122604148882, −24.26418193226319800363495286162, −23.08058983790223978170185926005, −21.2851923221573890229629011765, −20.20423138395625298963682816932, −19.02574812729203055218465338960, −17.85188192476284325873394535468, −16.563774671691519227382394855623, −15.21978777055287923460195806350, −14.48727094924965266602020644053, −13.263286813045280611520812577951, −10.88266487502327322643253360844, −9.74243815770659510254693360076, −8.28417889980620217857127900880, −7.66162290265441259641387734479, −5.65811874086761071458690937887, −3.99873869803303770716218160583, −1.59920888992399799804775150087, 1.419626679528417059099578114665, 2.79582447961465225648512508179, 4.447523573447786564205406489409, 7.16672996654210566574848130846, 8.44725429010390591402632294352, 9.29054917863484728887706647753, 10.94001723346847809653248257638, 12.18477963449148910679315618431, 13.51951634149810576751047022725, 14.66329475080473583535113728002, 16.3756901364290351407299021380, 18.04409639426043710236950385413, 18.67787599193744499373687612103, 19.984671452173336621493725002773, 20.88897698991125651230020479348, 21.77379946917152715632509496160, 23.671756902306521552329232717196, 24.991510557120913753123156737324, 26.02445554084137276827871358272, 27.04842864704104213174069278689, 28.08385535402607753066178440833, 29.40809437569849366886665774400, 30.576122163968513196531127377111, 31.070255736441503795803184987521, 32.29579714592882237453125762347

Graph of the $Z$-function along the critical line