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Properties

Label 1-21e2-441.317-r1-0-0
Degree $1$
Conductor $441$
Sign $0.0142 + 0.999i$
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  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.900 + 0.433i)5-s + (0.900 + 0.433i)8-s + (0.0747 − 0.997i)10-s + (−0.623 + 0.781i)11-s + (0.365 + 0.930i)13-s + (0.0747 − 0.997i)16-s + (0.733 + 0.680i)17-s + (−0.5 − 0.866i)19-s + (−0.955 + 0.294i)20-s + (0.955 + 0.294i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (0.733 − 0.680i)26-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.900 + 0.433i)5-s + (0.900 + 0.433i)8-s + (0.0747 − 0.997i)10-s + (−0.623 + 0.781i)11-s + (0.365 + 0.930i)13-s + (0.0747 − 0.997i)16-s + (0.733 + 0.680i)17-s + (−0.5 − 0.866i)19-s + (−0.955 + 0.294i)20-s + (0.955 + 0.294i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (0.733 − 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0142 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0142 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0142 + 0.999i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.0142 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7597059875 + 0.7489583931i\)
\(L(\frac12)\) \(\approx\) \(0.7597059875 + 0.7489583931i\)
\(L(1)\) \(\approx\) \(0.8761271263 - 0.06113590719i\)
\(L(1)\) \(\approx\) \(0.8761271263 - 0.06113590719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (-0.365 - 0.930i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (-0.826 - 0.563i)T \)
61 \( 1 + (-0.733 - 0.680i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.365 + 0.930i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
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

−23.82785685762493573503588055970, −22.92997713656758917824659553800, −22.09853738046764374080604523029, −20.99768030897854700845026885880, −20.31911916006999662720478950787, −18.88336571474307112320210416910, −18.39535227330741996690821978653, −17.48500563773038289260888077150, −16.59695752648470077263062335265, −16.09179496565504544669135054523, −14.91629665585529927675568364655, −14.086694540097595444878838369151, −13.28270848558385808380032898181, −12.50998578623311291628946077485, −10.77048637850726214371464813039, −10.161528401504265851204687898675, −9.10788946536307906464704698282, −8.329696146621760982559489882423, −7.44460985695949775451964180952, −6.03901811725660271833644368366, −5.658764338894891222981338765686, −4.59938707649155419296395283230, −3.04338717979183593547079844003, −1.460416125992267805625106904630, −0.32147656811237745509775780310, 1.4871211044785222791246484037, 2.2313569483513988056350176605, 3.36948236237831665806791480991, 4.54935780675284556401306890570, 5.67328147618922449208452236184, 6.96198479384294567224579020019, 7.96524039947209432464030082497, 9.27349526888905108979311692194, 9.71703605148381909159753439926, 10.78321178853556697444613135509, 11.4180146055756099563699460184, 12.74168919739535072798075120830, 13.254722255561435505223876680090, 14.24975537190340284586156303075, 15.19781370541163826320519889495, 16.66794372592558292952398268739, 17.304814000586291117589838119, 18.22479298024004848083167489239, 18.778795833884555597221077718341, 19.76840531392523569799960206718, 20.75517652918630865177138454129, 21.442613809058699302922542949209, 21.97813022475566548982814457731, 23.08757665734100044811289094474, 23.80417412876890305930294069342

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