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Properties

Label 1-21e2-441.421-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.944 + 0.328i$
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  + (0.365 − 0.930i)2-s + (−0.733 − 0.680i)4-s + (0.0747 − 0.997i)5-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)10-s + (0.365 − 0.930i)11-s + (−0.988 + 0.149i)13-s + (0.0747 + 0.997i)16-s + (−0.222 − 0.974i)17-s + 19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)26-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (−0.733 − 0.680i)4-s + (0.0747 − 0.997i)5-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)10-s + (0.365 − 0.930i)11-s + (−0.988 + 0.149i)13-s + (0.0747 + 0.997i)16-s + (−0.222 − 0.974i)17-s + 19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.944 + 0.328i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1746816274 - 1.033675420i\)
\(L(\frac12)\) \(\approx\) \(-0.1746816274 - 1.033675420i\)
\(L(1)\) \(\approx\) \(0.6469327873 - 0.7892277685i\)
\(L(1)\) \(\approx\) \(0.6469327873 - 0.7892277685i\)

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 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (-0.733 + 0.680i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (0.0747 + 0.997i)T \)
47 \( 1 + (0.365 - 0.930i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
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 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.988 - 0.149i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−24.43149330654233151595052601727, −23.794723831742040784001543030649, −22.58545159626929412693549666044, −22.36538055880878941395995149387, −21.50803229074946821149432848111, −20.25659467336353323556116321937, −19.22797244439228884998893799940, −18.23100919293364002094343489075, −17.51820527063934659762559987927, −16.840655704769162575581039677348, −15.4989633473220251060131460656, −15.02452735622854038616954613943, −14.26762619486831315835534723222, −13.367040226824927157442747356515, −12.32464665141372619949809899954, −11.4619110556134249723576943573, −10.00415798875657514998778362966, −9.457959391995327752004316590555, −7.91788522631816849719678060531, −7.34631774815741734154093036173, −6.42455242543440210775413199361, −5.507281204133779648746489000265, −4.315992121815807151158520717919, −3.37250389154607994679385152155, −2.09262158218201882934393047399, 0.516458658469297684226994088174, 1.75687275722689466011271447268, 2.975166476944110360766656861869, 4.106899739773066911446711857649, 5.08006079694338624635519419224, 5.786908517682648705152559594090, 7.333434069495798061566187507915, 8.73090693762941341508725189072, 9.262820503382432602824047450506, 10.247138726268978995877411334125, 11.41469396041919088869231821379, 12.09130616653724407823508261360, 12.86738721082742133615784945845, 13.890393292393691191235216593200, 14.4104174230259993653886452653, 15.85978545989790163419480348396, 16.61133971045696239159117903378, 17.70570386510855153748795449578, 18.55331063996036425990639689936, 19.62984726304488061336921286208, 20.13692107462709363348341495740, 20.95845468564544666933679768586, 21.87784170760376334836514801303, 22.42681282856406370605953587987, 23.59812851180920152094318151024

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