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Properties

Label 1-21e2-441.346-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.824 - 0.566i$
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  + (0.955 − 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

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

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.824 - 0.566i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.824 - 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3711291685 - 1.195607175i\)
\(L(\frac12)\) \(\approx\) \(0.3711291685 - 1.195607175i\)
\(L(1)\) \(\approx\) \(1.317997887 - 0.2864175172i\)
\(L(1)\) \(\approx\) \(1.317997887 - 0.2864175172i\)

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

−24.25199897483882770026485953655, −23.47409829709664558469990448368, −22.493268349744837758079542723289, −21.72377021052342859880044882152, −20.92314843103966306310536645997, −19.91125924875646382958014946829, −19.491629060615748918474990810284, −17.99439443328327467946685069740, −16.890972914678337032844339076889, −16.24379430558017981988758804256, −15.521111561583607771279129201262, −14.57723091849285933191879820405, −13.63754108337481526478081727551, −12.77565590208799414395927491156, −12.033438085264696783128723272282, −11.251176040277237179027042127, −10.08299332738420515486721838448, −8.54965234573103437613898556530, −7.980638340040671641350988653692, −6.88490725090194246324174172472, −5.70597872105962057516516439329, −4.93762288624505453573298623715, −3.91806465302329723564534366596, −2.993971233269836887403597390812, −1.514073278296798712096893798470, 0.21618682090755937800637069308, 2.175333066166960540777927868753, 2.79453013368848886322387438518, 4.20070043995485947755170593933, 4.740838959206092400471300479132, 6.16167053995466495262859483418, 7.089909863998218615804690825156, 7.71183852167352858025570291481, 9.50678657769651485046055202530, 10.29761317105969322914622249617, 11.3954958430316965784383578275, 11.90382638431445826069597195348, 12.91828742719754106696008782904, 13.91510668359441356657515153524, 14.75156881480124080931763253675, 15.421125778759881354122488979995, 16.15236122729129618282036356687, 17.545488981222786453705699909388, 18.46002157167710005629900332612, 19.58210672237601060313862502261, 19.97214131451415928681865279542, 21.012231165443841911870797400966, 22.10366018127763369504325503792, 22.528825265615462266887210308223, 23.30809054525295485117322284824

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