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Properties

Label 1-21e2-441.401-r1-0-0
Degree $1$
Conductor $441$
Sign $0.816 - 0.577i$
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  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.955 − 0.294i)5-s + (0.222 + 0.974i)8-s + (−0.733 − 0.680i)10-s + (−0.0747 − 0.997i)11-s + (0.0747 + 0.997i)13-s + (−0.222 + 0.974i)16-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.365 + 0.930i)26-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.955 − 0.294i)5-s + (0.222 + 0.974i)8-s + (−0.733 − 0.680i)10-s + (−0.0747 − 0.997i)11-s + (0.0747 + 0.997i)13-s + (−0.222 + 0.974i)16-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.365 + 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.295961227 - 0.7306627269i\)
\(L(\frac12)\) \(\approx\) \(2.295961227 - 0.7306627269i\)
\(L(1)\) \(\approx\) \(1.485373603 + 0.1487101625i\)
\(L(1)\) \(\approx\) \(1.485373603 + 0.1487101625i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.955 - 0.294i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.365 - 0.930i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.0747 + 0.997i)T \)
89 \( 1 + (-0.826 - 0.563i)T \)
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

−23.58914757118794137223944585856, −22.99372465452937703982378786503, −22.477000120574836314043391517286, −21.36453292264801909219094595478, −20.48825231296745816313535678754, −19.787578964751334926964409256354, −19.077250916082884187475301920, −18.07997175558821732952549352011, −16.86405345869642558951466718285, −15.67330878513778036539719071618, −15.08849432814632063039003685304, −14.52357212774335919271500471988, −13.10389111586834087714058170077, −12.55853681582105002270755078413, −11.699582052352286039753280631601, −10.674021425962615949507194723160, −10.1345344439987870324406999581, −8.56866718639165702773171375660, −7.46300017441907822919833640370, −6.614700093624861160238908839713, −5.41938700556501886369058340978, −4.38790439192084136588116122669, −3.56733538817506798551368567632, −2.52954047175908668219110372638, −1.16513911698032395938479162037, 0.50544572661939621470238414757, 2.39305632112506269176774118463, 3.4984975576080343246284048547, 4.41368421001834964988053000352, 5.234347513302657022436789551643, 6.54483052411595647884715861156, 7.243280597197032491622524089971, 8.36972038646294023156620662700, 9.07150970000814804063342114496, 10.953099294501914482643600592084, 11.480605973648637243178706126841, 12.314021577102608154829701119208, 13.40473435360021601088082717046, 14.003892035845427764049257003600, 15.19844888417428999865266789376, 15.78792096269080366435191168603, 16.575180177549010471698803773681, 17.34685896285100264135396531935, 18.85423642083232123147864965503, 19.459609298863028524147358905031, 20.625694562519508343073601670153, 21.20967249925634616099615395606, 22.22265603119314123119987193341, 23.00239957929966093262161043494, 23.82941934919608776765109540466

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