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Properties

Label 1-21e2-441.88-r0-0-0
Degree $1$
Conductor $441$
Sign $0.430 - 0.902i$
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  + (−0.222 − 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

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

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.430 - 0.902i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078524845 - 0.6803642328i\)
\(L(\frac12)\) \(\approx\) \(1.078524845 - 0.6803642328i\)
\(L(1)\) \(\approx\) \(0.9293177395 - 0.4665857671i\)
\(L(1)\) \(\approx\) \(0.9293177395 - 0.4665857671i\)

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

−24.418381893326993251195928447534, −23.24354372489981488773234938346, −22.70526164001530358811074749812, −21.95645178592288253164888113028, −20.93422708756531115840671396228, −19.58339668202495831109083008562, −18.81114986812252541340792824400, −18.09150084431808470046543125875, −17.28873665700007477986225749137, −16.46196663261835381048499567624, −15.37120640112898771392009401650, −14.81398087534774868559543323533, −13.74196071822504258471463830666, −13.345846640727924437079218069460, −11.69043763841102785388214023393, −10.767619732777142823966573097465, −9.763292969865773640374025212022, −8.918892343838653573301299715324, −7.93859901131752862844125200160, −6.65090151126309694596765505690, −6.44210802279498852384952981548, −5.16132328865290821377063771287, −3.97857029674979354240196885067, −2.76635275768859924235028729082, −1.03366962718955399239824979823, 1.21472431801784324147653515687, 1.84358319565128165090325466849, 3.505932728850079756063270884819, 4.26696368234296859117300429063, 5.376437156495071812024686563241, 6.54775677078679872297989446738, 8.17773106281963165909456333461, 8.7736052117806921912449176225, 9.62854522613829332030980388839, 10.547079099877621190693270327734, 11.60196099530654554216614179644, 12.39829888486255331624711052342, 13.159936361413258831003855506286, 13.972608351670297634021536297583, 15.055246969990743015874145711307, 16.55524735250601949635086236104, 17.00176905750009085223234875418, 17.92074471841603834017884671591, 18.89815262671151139017159893915, 19.75095833284061194074207929952, 20.43945209798434395299306046296, 21.3263978113851914047230815959, 21.79266692307006839010796340885, 23.08009382739730833072429885240, 23.61745447811989393191475195093

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