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Properties

Label 1-21e2-441.142-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.713 - 0.700i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
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  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (0.623 − 0.781i)5-s + (0.623 − 0.781i)8-s + (−0.988 + 0.149i)10-s + (−0.222 + 0.974i)11-s + (−0.733 − 0.680i)13-s + (−0.988 + 0.149i)16-s + (0.0747 − 0.997i)17-s + (−0.5 − 0.866i)19-s + (0.826 + 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (0.623 − 0.781i)5-s + (0.623 − 0.781i)8-s + (−0.988 + 0.149i)10-s + (−0.222 + 0.974i)11-s + (−0.733 − 0.680i)13-s + (−0.988 + 0.149i)16-s + (0.0747 − 0.997i)17-s + (−0.5 − 0.866i)19-s + (0.826 + 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.713 - 0.700i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2884289614 - 0.7051934675i\)
\(L(\frac12)\) \(\approx\) \(0.2884289614 - 0.7051934675i\)
\(L(1)\) \(\approx\) \(0.6355435983 - 0.3810301369i\)
\(L(1)\) \(\approx\) \(0.6355435983 - 0.3810301369i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.733 - 0.680i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (0.365 + 0.930i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (0.365 - 0.930i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
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

−24.46642483051508346845137574145, −23.74021647191187060823089988832, −22.83415960271365376699415882940, −21.73649110136935797036089717769, −21.15392196719503111948462247977, −19.57626547467002194964438219904, −19.1788805164426788248478672417, −18.23108961568147854167066871328, −17.506583822624902146674245205013, −16.65288443211180301700056369153, −15.86759419510167182030205091378, −14.61793080407588414893468210171, −14.315814557828951390070479945946, −13.254614662738648186726842388266, −11.80515617660744839247536173846, −10.695854342397542276488452267113, −10.15184731772223975378534092282, −9.14973998917729404453427894354, −8.159225187304466533278047918081, −7.23414511285693141058805648374, −6.153842314059583431218015576039, −5.7055596210440825690614988132, −4.15124661069224849252990725437, −2.60628085615636559189889934828, −1.53294647413041953138297918533, 0.543208271229566320318738214875, 1.98271689523981847972659951061, 2.74204373232383234220521549931, 4.353160789940360038057649167512, 5.13994408684880320668171489002, 6.67494369843404619153964292893, 7.71112532709838935599123021724, 8.61918652560996218440544088295, 9.69056637125352671930060563055, 10.02677949360785814529613228223, 11.293837403711641622847727173153, 12.35240919536090387734005130602, 12.850369381138297626344477301529, 13.82753870166484414642523848283, 15.16434936371602905556290423343, 16.22366022645732849333842615351, 17.01582279337093612912099604905, 17.83123386971847355205338430532, 18.32484735702503369669827313836, 19.79773773609407071159288553130, 20.13886498981151008081935407702, 20.96006263070136522869676595587, 21.86019955221654504412435288061, 22.57766451610910955258220016927, 23.8489303034447418300583605763

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