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Properties

Label 1-55-55.53-r1-0-0
Degree $1$
Conductor $55$
Sign $0.486 + 0.873i$
Analytic cond. $5.91057$
Root an. cond. $5.91057$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(5.91057\)
Root analytic conductor: \(5.91057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (1:\ ),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.492593171 + 0.8775273885i\)
\(L(\frac12)\) \(\approx\) \(1.492593171 + 0.8775273885i\)
\(L(1)\) \(\approx\) \(1.134410250 + 0.4760026104i\)
\(L(1)\) \(\approx\) \(1.134410250 + 0.4760026104i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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