Properties

Label 1-21e2-441.389-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.968 - 0.250i$
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 + (0.988 + 0.149i)5-s + (−0.623 + 0.781i)8-s + (0.365 − 0.930i)10-s + (0.733 + 0.680i)11-s + (−0.733 − 0.680i)13-s + (0.623 + 0.781i)16-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.826 + 0.563i)26-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.988 + 0.149i)5-s + (−0.623 + 0.781i)8-s + (0.365 − 0.930i)10-s + (0.733 + 0.680i)11-s + (−0.733 − 0.680i)13-s + (0.623 + 0.781i)16-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.826 + 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.968 - 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2074576696 - 1.631911286i\)
\(L(\frac12)\) \(\approx\) \(0.2074576696 - 1.631911286i\)
\(L(1)\) \(\approx\) \(0.9349462652 - 0.6766011347i\)
\(L(1)\) \(\approx\) \(0.9349462652 - 0.6766011347i\)

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 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (-0.0747 - 0.997i)T \)
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 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + T \)
83 \( 1 + (0.733 - 0.680i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
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

−24.32277313274404403603882173432, −23.69825906596391634212398958278, −22.37717023859546220986387819821, −21.80754661985293733452897510235, −21.29304139524228616664547659727, −19.83357316411545795474824953331, −18.935963540119081560671406790, −17.85345117083957324529534002816, −17.13571059679602925161838391098, −16.664331324984846278007650198030, −15.463348787529593404872826700165, −14.63565883854684474232032233012, −13.75246800247620356599757116415, −13.21807535424795914495241068572, −12.11495087461263318777418866490, −10.90748570504131244978315796450, −9.477186299625908195421853968882, −9.12710693679998673159958187495, −7.99166506238206420222439585043, −6.681399929972194634581908372360, −6.2384717413700075063201126560, −5.07553549229318449459884493901, −4.21966980515275858591753463708, −2.81808730546581757522474232275, −1.32377625052034750581588361120, 0.40197753434844885971022566274, 1.89057576938339259619043843394, 2.50171369482548149916995980175, 3.8909077968744775704997362178, 4.90329477183872486878776495831, 5.87238048563920663032671480497, 6.95826454873542325066013483220, 8.4693690337018011464900134622, 9.45169302544274598940635425947, 10.094860792270257221491960596964, 10.91060003040461795383291219231, 12.12033382325717128825070660981, 12.733558847353350871167699221928, 13.731409424720943091468406721940, 14.466725727171357618512321070056, 15.25644179321327839963311266076, 16.93444488974380795113816817476, 17.51777454502510011446975204179, 18.34258909768626882989614903658, 19.23047917124777370495354004828, 20.27191245779179376703748572781, 20.746767932013843622940236739993, 21.79480256624637741156863230935, 22.49105491935295265256860998651, 22.958709561748009087884498280195

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