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Properties

Label 1-21e2-441.122-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.995 + 0.0924i$
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  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.222 − 0.974i)5-s + (0.222 + 0.974i)8-s + (−0.955 + 0.294i)10-s + (0.900 + 0.433i)11-s + (−0.0747 − 0.997i)13-s + (0.955 − 0.294i)16-s + (−0.988 − 0.149i)17-s + (0.5 − 0.866i)19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.222 − 0.974i)5-s + (0.222 + 0.974i)8-s + (−0.955 + 0.294i)10-s + (0.900 + 0.433i)11-s + (−0.0747 − 0.997i)13-s + (0.955 − 0.294i)16-s + (−0.988 − 0.149i)17-s + (0.5 − 0.866i)19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.995 + 0.0924i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.995 + 0.0924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03812696273 - 0.8228051880i\)
\(L(\frac12)\) \(\approx\) \(-0.03812696273 - 0.8228051880i\)
\(L(1)\) \(\approx\) \(0.5768233457 - 0.5954632569i\)
\(L(1)\) \(\approx\) \(0.5768233457 - 0.5954632569i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.0747 - 0.997i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.733 + 0.680i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (-0.365 + 0.930i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.0747 - 0.997i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
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

−24.514412459711581690013747354930, −23.7029263690995562734666072325, −22.90252890016783556841315857594, −22.03805279183265511591375787655, −21.58184303085459178913674415586, −19.86782885414668593429286603435, −19.16891968953083712739236313326, −18.32342959565150909427392688023, −17.60502109606227995451599943248, −16.5594251193210867013460801028, −15.87313844786767283847127787121, −14.84315336064386268200025052159, −14.20498771508338175577432192675, −13.55371986146599124179811507392, −12.132386158245779309801661751485, −11.22293945955016786852667637039, −10.08343918367630418722155026684, −9.16072681262942630589945946663, −8.23346046412481259155643703696, −7.071864515587082524593698922044, −6.58326894893738946587439404477, −5.56106166235715655978983979689, −4.18020980028645722242138700316, −3.44845316285908514026534489295, −1.67909390090680288085895675527, 0.49856735799824700175451222567, 1.72232210373847198314550486960, 2.94288970983060196146831193761, 4.2549766790407930247817250161, 4.78548038058866634690918194228, 6.09405436101554760307643964110, 7.65954825320705211990482879717, 8.58137233412939544548718009768, 9.36997842675323200897944234513, 10.193324440648855174730272105768, 11.4666110838713452306870953107, 11.976788702160206725107629314, 13.02961525664443716634586624839, 13.52919718164303984512997283470, 14.83313747755885969738838467853, 15.77709488109230844739858176758, 17.07934428803713187334356039039, 17.50189192379803591923758286287, 18.571343122769971099666355682290, 19.65766199710217349170142282425, 20.24539158158391749032505461748, 20.66806908229631580353956835132, 22.075841302283155770561224986955, 22.385228195845470738154351853273, 23.5232930687874958620910107922

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