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Properties

Label 1-21e2-441.220-r1-0-0
Degree $1$
Conductor $441$
Sign $0.0924 + 0.995i$
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  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.222 + 0.974i)5-s + (−0.222 − 0.974i)8-s + (−0.955 + 0.294i)10-s + (−0.900 − 0.433i)11-s + (−0.0747 − 0.997i)13-s + (0.955 − 0.294i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (0.623 + 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.988 − 0.149i)26-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.222 + 0.974i)5-s + (−0.222 − 0.974i)8-s + (−0.955 + 0.294i)10-s + (−0.900 − 0.433i)11-s + (−0.0747 − 0.997i)13-s + (0.955 − 0.294i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (0.623 + 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.988 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0924 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0924 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0924 + 0.995i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.0924 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.327816559 + 1.210210050i\)
\(L(\frac12)\) \(\approx\) \(1.327816559 + 1.210210050i\)
\(L(1)\) \(\approx\) \(0.9035649244 + 0.5711599189i\)
\(L(1)\) \(\approx\) \(0.9035649244 + 0.5711599189i\)

Euler product

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

−23.34159666534175357879207635597, −22.92619382763796006196591822561, −21.47971952409999358628029000034, −21.08043950696577472213006845682, −20.4227001383148338957828084904, −19.42434903131816941030975439217, −18.62690033618510995841930545489, −17.76602212942312382924852016101, −16.78658127970895351180013020528, −16.00785144309602242405806412827, −14.57228694682502021527859858368, −13.833656127681359701998017023198, −12.81340219319884627941857204142, −12.287489893612225161671729733197, −11.36563239372349019101353337173, −10.1377333633097722901421765193, −9.5789601800883385919061498261, −8.551108129093238274377927347480, −7.67681002761423470238467588595, −5.96982181950402267254031093469, −4.97060790061973610169861618259, −4.27424555547428514166923969703, −2.9211139063566514249463914570, −1.79746769152004852068500920948, −0.76266867168243778171997118830, 0.733688623308017294607882641643, 2.782794138178705270288739256603, 3.55785185847799281074150746725, 5.1797636149337819228869769855, 5.71539375676873541025696590609, 6.92191651407940653743149341329, 7.61978258381080395426218097474, 8.52634180989376933191675895322, 9.822851688161480235347968807637, 10.453892965832623494397318588719, 11.67269689486810099261353382797, 13.064027681501128865654807432497, 13.57995584168136965034118382659, 14.65467043180384158471862920855, 15.28569943789931245888226957815, 16.066238343715879326297949164045, 17.18464416154423832999865092446, 17.94077358055582834686309205186, 18.60880241086567237417354558772, 19.44103476975271372505897814107, 20.86347594763409945264171598675, 21.78218418475995584532470539235, 22.44768750243232237357017340730, 23.31724476062967494523559852873, 23.93508525830737950616793058653

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