Properties

Label 1-21e2-441.137-r1-0-0
Degree $1$
Conductor $441$
Sign $0.902 + 0.430i$
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  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (−0.955 − 0.294i)11-s + (0.955 + 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (−0.955 − 0.294i)11-s + (0.955 + 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.902 + 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132295912 + 0.2563183997i\)
\(L(\frac12)\) \(\approx\) \(1.132295912 + 0.2563183997i\)
\(L(1)\) \(\approx\) \(0.7793629759 + 0.4671539131i\)
\(L(1)\) \(\approx\) \(0.7793629759 + 0.4671539131i\)

Euler product

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

−23.562482385007702719644440934394, −23.105299594734756455636617627982, −21.72830985757977063197882824059, −21.2466024876846389280326898450, −20.22392016070024452488672322352, −19.839406517659747289881216837425, −18.74389479751442310442248373637, −17.910474597406292013889501030561, −17.03595891623208901471921243568, −15.782337140531416523899946000530, −15.13717807982879079356506286473, −13.74802277830164648884670260209, −13.02929926988905576951547001458, −12.46236738250936008295226448798, −11.374345663408875253725093558864, −10.602961036665868454572542944963, −9.60020479402691570896035796305, −8.553338536666383259158803133480, −7.96373203617830434565184226412, −6.17178765155261327535454204807, −5.13545516931713182498685735689, −4.30940366546403867622903323409, −3.29974572841285963538211069986, −1.98406534089324435495415195998, −0.87927448331091698686413637376, 0.37571093378805575063284728542, 2.52869247042109340705812480100, 3.630953177307524351820178079884, 4.57609284478797378523186022624, 5.93312740972929090656814458166, 6.49337496534861800684588129808, 7.73642578287805365461829627468, 8.182959375755009237484470771664, 9.514489679030318025677153256205, 10.5255852401135894714668004781, 11.54169127000911544319604756696, 12.648629402089821153247791556527, 13.75853557494317054111161854413, 14.22885328578308070200918620488, 15.38860880813427876521658399574, 15.884152880987286341371232330779, 16.75059360273982361354422502721, 18.08920920115111539728031196106, 18.40407211686006416453052632979, 19.26528655699197540990270080508, 20.76573269523989820212875771225, 21.45912888913444807527216148968, 22.6052289376681093297592314179, 23.05129877288982679404017677969, 23.79429149799859046009647958598

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