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Properties

Label 1-21e2-441.94-r1-0-0
Degree $1$
Conductor $441$
Sign $0.328 - 0.944i$
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  + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (0.900 + 0.433i)5-s + (−0.900 − 0.433i)8-s + (−0.826 − 0.563i)10-s + (0.623 − 0.781i)11-s + (0.988 + 0.149i)13-s + (0.826 + 0.563i)16-s + (−0.955 + 0.294i)17-s + (0.5 − 0.866i)19-s + (0.733 + 0.680i)20-s + (−0.733 + 0.680i)22-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.955 − 0.294i)26-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (0.900 + 0.433i)5-s + (−0.900 − 0.433i)8-s + (−0.826 − 0.563i)10-s + (0.623 − 0.781i)11-s + (0.988 + 0.149i)13-s + (0.826 + 0.563i)16-s + (−0.955 + 0.294i)17-s + (0.5 − 0.866i)19-s + (0.733 + 0.680i)20-s + (−0.733 + 0.680i)22-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.955 − 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179800188 - 0.8386933648i\)
\(L(\frac12)\) \(\approx\) \(1.179800188 - 0.8386933648i\)
\(L(1)\) \(\approx\) \(0.8720573313 - 0.1490944117i\)
\(L(1)\) \(\approx\) \(0.8720573313 - 0.1490944117i\)

Euler product

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

−24.360335329222487335135816388186, −23.31091929999412363984530765981, −22.21599820473936488086863955282, −21.17016009426374141526410269044, −20.40835066470009304372641582767, −19.839977724082120978304135940424, −18.58844242391982463804585087020, −17.87361631276498624732469296436, −17.294712396954161456092919104720, −16.33814317554791907347487837950, −15.58927196521584509996285514524, −14.490996142389706818459096843697, −13.54240289278436283609101073032, −12.438565477477160188854152589004, −11.46240285003247789926359762109, −10.44639390941182078221189355031, −9.55851398942189195652391001405, −8.96231557408926395514488958695, −7.93996590377157046847845150451, −6.76410492465023224442991582340, −6.028884496643820303167534633241, −4.918902781792805256592593861812, −3.32507092571707680126502125898, −1.85700778534844954348673382999, −1.237707313447075255534297144977, 0.5528685373620678467800695624, 1.79177373936028555421069681237, 2.7595290732802542777708122832, 3.97300787397826556623123530202, 5.83069716900845439122446995379, 6.40787334114771897669444850227, 7.39817191813029084463154736912, 8.836708513858406107030706412323, 9.080910096326634325579219787, 10.41181322107905434625077610787, 10.973863625424776921064268586170, 11.87461344623309652821106755164, 13.22546306813496418783037789480, 13.950639082642401853918122082402, 15.16471439551578976367594395940, 16.01201926137955916510144791227, 17.03701591583788473558208948191, 17.60400678740703416479704356478, 18.56036614593756530702573254051, 19.09880954395611789791916127753, 20.25599073411306364414902750953, 20.94433861088266337999899304372, 21.88658871784073291692357882786, 22.51221675811575035684741118356, 24.132529619785249668418433596415

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