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Properties

Label 1-21e2-441.268-r0-0-0
Degree $1$
Conductor $441$
Sign $0.999 + 0.0142i$
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  + (0.365 + 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.900 − 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.0747 − 0.997i)10-s + (0.623 − 0.781i)11-s + (0.365 + 0.930i)13-s + (0.0747 − 0.997i)16-s + (−0.733 − 0.680i)17-s + (−0.5 − 0.866i)19-s + (0.955 − 0.294i)20-s + (0.955 + 0.294i)22-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.733 + 0.680i)26-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.900 − 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.0747 − 0.997i)10-s + (0.623 − 0.781i)11-s + (0.365 + 0.930i)13-s + (0.0747 − 0.997i)16-s + (−0.733 − 0.680i)17-s + (−0.5 − 0.866i)19-s + (0.955 − 0.294i)20-s + (0.955 + 0.294i)22-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.733 + 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0142i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.999 + 0.0142i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054650633 + 0.007513239753i\)
\(L(\frac12)\) \(\approx\) \(1.054650633 + 0.007513239753i\)
\(L(1)\) \(\approx\) \(0.9423004921 + 0.2603904020i\)
\(L(1)\) \(\approx\) \(0.9423004921 + 0.2603904020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.365 + 0.930i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.955 - 0.294i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (0.826 - 0.563i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (0.955 + 0.294i)T \)
59 \( 1 + (0.826 + 0.563i)T \)
61 \( 1 + (-0.733 - 0.680i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
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

−23.54256121804822566531185926222, −23.20531484967475764035617969764, −22.317319824169941697303808390436, −21.60609973684829152838222942002, −20.43329046628195114206443976442, −19.81156891771777761185562535083, −19.228746168955726763509330424214, −18.121557387341076942364669744757, −17.52098458662453742165853305897, −16.02172688759276438230571290401, −15.03751508607473397364409588988, −14.56801496392720610271428984646, −13.32240969205455452833411086006, −12.447745175712483747909014100231, −11.77137294362767333774457052202, −10.77587016127272212871084427695, −10.16307882184793244842551656619, −8.909229979895223153514855429463, −7.992998882797738014161032473959, −6.73619297039492561424310702904, −5.62166590837701770390529918299, −4.29438404424193059256140538952, −3.71798220174239509264929352407, −2.56913508659756624488263670740, −1.264027655100986308459354253095, 0.61147512290570468832534893799, 2.75887729144551046593108413642, 4.18552488559391476661715072077, 4.462445557153488672438855272048, 5.95175859086214077458024121438, 6.774625324293181529225613008550, 7.736254093877165309391745955413, 8.80205604673566687333881972406, 9.17276925608914843127559191858, 11.04863325779220108420476434466, 11.78392568740756664560643491590, 12.76701651575495496649028489612, 13.688193649539976125687967633438, 14.49530588337035575919009432388, 15.49004374627474937398170223208, 16.23354350322214056149408043030, 16.76563786953947458287478409988, 17.86270692426933700339475649811, 18.86424630917614099498204177455, 19.65000330306182122953129884803, 20.7841432527115339738310971777, 21.69986004584339146152155375918, 22.53705257277855697421750841795, 23.34496788507438627818450147311, 24.28246620695590916186817222295

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