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Properties

Label 1-21e2-441.113-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.908 - 0.417i$
Analytic cond. $47.3920$
Root an. cond. $47.3920$
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  + (−0.826 + 0.563i)2-s + (0.365 − 0.930i)4-s + (0.733 − 0.680i)5-s + (0.222 + 0.974i)8-s + (−0.222 + 0.974i)10-s + (−0.826 + 0.563i)11-s + (0.0747 + 0.997i)13-s + (−0.733 − 0.680i)16-s + (−0.623 + 0.781i)17-s + 19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.623 − 0.781i)26-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)2-s + (0.365 − 0.930i)4-s + (0.733 − 0.680i)5-s + (0.222 + 0.974i)8-s + (−0.222 + 0.974i)10-s + (−0.826 + 0.563i)11-s + (0.0747 + 0.997i)13-s + (−0.733 − 0.680i)16-s + (−0.623 + 0.781i)17-s + 19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.623 − 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.908 - 0.417i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001027736646 + 0.004695142593i\)
\(L(\frac12)\) \(\approx\) \(0.001027736646 + 0.004695142593i\)
\(L(1)\) \(\approx\) \(0.6530304452 + 0.1075316881i\)
\(L(1)\) \(\approx\) \(0.6530304452 + 0.1075316881i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.826 + 0.563i)T \)
5 \( 1 + (0.733 - 0.680i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (-0.955 + 0.294i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.0747 + 0.997i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−24.6362194743922738075420703604, −23.16258844587443312774442896764, −22.14563101641364896437146499679, −21.699222386716309643689117609086, −20.534896095009135241343684241966, −20.09740246075923891775984439450, −18.77270691645029248276037928304, −18.1635384457345419514534208033, −17.74963617676216496709130987422, −16.47403957372920404125135636174, −15.809426643916504215101787507869, −14.584729547079490654450337629227, −13.456616037272311202046216978481, −12.82287729758326480809218121161, −11.527138734679888022611339812210, −10.74385075217757744867158692519, −10.08662928026900937906727067142, −9.16225538316194015428705912982, −8.10485792780947924193379551778, −7.21416896515947948656340177660, −6.16250785030045402028663396706, −4.96234309905548806731137394484, −3.17388966228027359874089261342, −2.752634886993504060262954858011, −1.376879983664348893247509102363, 0.00162879809347638500471393780, 1.51538132891563695071609370780, 2.28228574588262880350622080836, 4.28526888820248599263808136115, 5.38674014332326315149102761080, 6.12557682395889987918572333540, 7.3011207130235640828516387418, 8.154449306828335272481347802406, 9.29477924992459096364488422978, 9.69415358312231846269726235158, 10.808495300782228774597057199665, 11.83826502668750594587691187582, 13.1217413166312067289524935147, 13.85738219671993632686985694305, 14.94420597055683523061986578021, 15.871407352621334679577316146406, 16.57494987715522305119441966899, 17.51170244050139690399449431183, 18.0286646473067759796687033733, 19.04042417881302637923613770110, 19.99134538216469772568511185572, 20.75686392418696434360093128722, 21.59218653172440768806279789656, 22.80644611891632808645902050568, 23.997800626869036399198275219311

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