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Properties

Label 1-21e2-441.353-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: $\chi_{441} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6252932396 - 0.3944528096i\)
\(L(\frac12)\) \(\approx\) \(0.6252932396 - 0.3944528096i\)
\(L(1)\) \(\approx\) \(0.8604196172 + 0.1299410290i\)
\(L(1)\) \(\approx\) \(0.8604196172 + 0.1299410290i\)

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

−23.86031338310949803377717461335, −23.20962129199325592787552689107, −22.16132152570010247919531519343, −21.864756867945154942163231440453, −20.73149746793687771791977651857, −20.08429266316454157808098891448, −18.97952601263364151668345955655, −18.30564085528741719883209596137, −17.77226463810416519946216911372, −16.46735734372566698306949080496, −15.19720843331161562675699841271, −14.36720205791445229618970770213, −13.72286275719320578191175215370, −12.69174335672006134324258022743, −11.78238444238420205565499147780, −10.940288857318043999970271544724, −9.93448993012597886929443689562, −9.580705066511191777489581534484, −8.01629543272886317310104800094, −7.04427210839647238629868719834, −5.67143002756551541899159330932, −4.89186183801589598911134851590, −3.512015562734708913579389216746, −2.66204074220734046676008009520, −1.73238126980852568928810313863, 0.36950104294953263523528797215, 2.257286959282478020470402926574, 3.75204723768111735259360424654, 4.92905238413509896354699141268, 5.467440395183873479560278788581, 6.54018603493233106988023054750, 7.80233985622099734658774274404, 8.35886563837529416073577131767, 9.45843264634375812208894174243, 10.23981611396420840410987351239, 11.82429889692022863953646631657, 12.8802846522484531030412778026, 13.2607171324832549315151704501, 14.37230290058502088766795149067, 15.28936763157807612031946589381, 16.12389476422318152721914290748, 16.883487609726909559442068151685, 17.62297311283613903569755092997, 18.434245547407243359236724429, 19.63656900449090676723314011823, 20.58474806042834720646114079316, 21.6653786818557467206436486218, 22.067984926430108532083549861952, 23.42405337318226339315020710387, 23.98967189587251034618070249282

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