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Properties

Label 1-21e2-441.205-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.822 - 0.569i$
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  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.222 − 0.974i)5-s + (−0.222 − 0.974i)8-s + (−0.733 − 0.680i)10-s + (−0.900 − 0.433i)11-s + (0.826 − 0.563i)13-s + (−0.733 − 0.680i)16-s + (0.365 + 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.623 + 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.222 − 0.974i)5-s + (−0.222 − 0.974i)8-s + (−0.733 − 0.680i)10-s + (−0.900 − 0.433i)11-s + (0.826 − 0.563i)13-s + (−0.733 − 0.680i)16-s + (0.365 + 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.623 + 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.365 − 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.822 - 0.569i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.822 - 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5320169930 - 1.703194210i\)
\(L(\frac12)\) \(\approx\) \(0.5320169930 - 1.703194210i\)
\(L(1)\) \(\approx\) \(1.130240513 - 0.9541029694i\)
\(L(1)\) \(\approx\) \(1.130240513 - 0.9541029694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.826 - 0.563i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (0.826 - 0.563i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.955 - 0.294i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
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

−24.24851503878615296667427559668, −23.28194037349320718425712924727, −22.93575451208693656588263506201, −22.0761515984871844974850978192, −20.960831373836592936056445348568, −20.60684944460756526588010152596, −19.03155315389305005674104159506, −18.37808451933637384757685733170, −17.43459207732756527176211989944, −16.21606207031260246543822331478, −15.73769968363557384984917156430, −14.62266172059453785288827202682, −14.17816868746615794324981489909, −13.088855741114801625245525966895, −12.2391215155092200796049013830, −11.1779224373442346178907469710, −10.48983623971700310595970788068, −9.01696177884901658446872191139, −7.83296024477246806955611594614, −7.13657868227454529832048855156, −6.238351339708286990853936173688, −5.23142850973583578468470362003, −4.075410018459206429751092162372, −3.1477117927420967444773145241, −2.136591276069495336200963682591, 0.75753461860332292186942585310, 2.014541558450534983085538249349, 3.345208487381529043468530056292, 4.20702395208300362840575727036, 5.38142716040848481778848352636, 5.8749192210846043572338151159, 7.41715213718849800027315814580, 8.514404776065617545313872155776, 9.48490460066387991784008081379, 10.72665199948546545736090636111, 11.267886669820093575901178885847, 12.54209757993314583001378861243, 13.04656622009586866274495608588, 13.71600546141059355863377541207, 15.11917822632154553323285235749, 15.62106396978982087019208661350, 16.590126176642272294398796295831, 17.69312707085949441303315608531, 18.917438440080809121094848927304, 19.51962393812600294336393624919, 20.62792066213361807885738689019, 20.9758929027031597816860790367, 21.87242778195535172262862249937, 22.95066068938832181856872353486, 23.785137570986666866977927732283

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