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Properties

Label 1-21e2-441.425-r0-0-0
Degree $1$
Conductor $441$
Sign $0.566 + 0.824i$
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  + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (0.623 + 0.781i)5-s + (−0.623 − 0.781i)8-s + (−0.365 − 0.930i)10-s + (0.222 + 0.974i)11-s + (−0.955 − 0.294i)13-s + (0.365 + 0.930i)16-s + (0.826 − 0.563i)17-s + (0.5 + 0.866i)19-s + (0.0747 + 0.997i)20-s + (0.0747 − 0.997i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.826 + 0.563i)26-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (0.623 + 0.781i)5-s + (−0.623 − 0.781i)8-s + (−0.365 − 0.930i)10-s + (0.222 + 0.974i)11-s + (−0.955 − 0.294i)13-s + (0.365 + 0.930i)16-s + (0.826 − 0.563i)17-s + (0.5 + 0.866i)19-s + (0.0747 + 0.997i)20-s + (0.0747 − 0.997i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.826 + 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.566 + 0.824i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.566 + 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8194712144 + 0.4312378379i\)
\(L(\frac12)\) \(\approx\) \(0.8194712144 + 0.4312378379i\)
\(L(1)\) \(\approx\) \(0.7897269310 + 0.1286051987i\)
\(L(1)\) \(\approx\) \(0.7897269310 + 0.1286051987i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.955 - 0.294i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.955 - 0.294i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (-0.0747 - 0.997i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 + (-0.988 + 0.149i)T \)
43 \( 1 + (-0.988 - 0.149i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (-0.988 - 0.149i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.955 - 0.294i)T \)
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

−24.22652861639935581146968127872, −23.47635456755271471970658437254, −21.9025602878796581407964990066, −21.320915335680002525785773277924, −20.30621757144730847141891527179, −19.52588707865131708815400277736, −18.77086143745161587468636559786, −17.71969041081790445688180750199, −16.89692598823321894482064008512, −16.5382195730750381514251991427, −15.39998318720508096057267208200, −14.42414104495919002777935880262, −13.4906240875100077456196789822, −12.31842459095345160438009524312, −11.40852253560647694473108968867, −10.35376836618971513574079657440, −9.42190488802436933268007075256, −8.83687412819939607384548815713, −7.8260351934684831449915193720, −6.781997523032451669921559880870, −5.71559265550032712533976541380, −4.96283039934351247556440490126, −3.17329949957233176640929206405, −1.88997752128501366189024880380, −0.77603481607106365922832494603, 1.37368538365456508679777057131, 2.49045442914480526444636298528, 3.31728691517003431593576860491, 4.97039109484059955357624521118, 6.29586599961774559547199533933, 7.17808876357334596709319104934, 7.88404087597505512335669607396, 9.29457553294259304993863828585, 9.962364676602202188776864841863, 10.523333797456384489156157883868, 11.81642396463940663737940966681, 12.38956314587583300277718931059, 13.716355614565437873794980885775, 14.78743968599620860537550303946, 15.44913457351384213920905589489, 16.878822360562495482748965176521, 17.231538569439492167794265585527, 18.33754659024882431998606647328, 18.763259277645977341667026462074, 19.85986073145316409336691431468, 20.65434819035511671204478736589, 21.45406108349385423839633760127, 22.41009376480059547230382881453, 23.11989085749517350315148189146, 24.71752775749964159262625071978

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