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Properties

Label 1-21e2-441.236-r0-0-0
Degree $1$
Conductor $441$
Sign $0.822 + 0.569i$
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  + (−0.826 + 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.222 − 0.974i)5-s + (0.222 + 0.974i)8-s + (0.733 + 0.680i)10-s + (0.900 + 0.433i)11-s + (−0.826 + 0.563i)13-s + (−0.733 − 0.680i)16-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.222 − 0.974i)5-s + (0.222 + 0.974i)8-s + (0.733 + 0.680i)10-s + (0.900 + 0.433i)11-s + (−0.826 + 0.563i)13-s + (−0.733 − 0.680i)16-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.365 − 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.822 + 0.569i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.822 + 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8112466471 + 0.2534044556i\)
\(L(\frac12)\) \(\approx\) \(0.8112466471 + 0.2534044556i\)
\(L(1)\) \(\approx\) \(0.7377406261 + 0.1216889453i\)
\(L(1)\) \(\approx\) \(0.7377406261 + 0.1216889453i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.826 + 0.563i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (0.826 - 0.563i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (0.955 - 0.294i)T \)
61 \( 1 + (-0.365 - 0.930i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.0747 + 0.997i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
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

−24.18386352063063838166340669754, −22.699048310577710049995248288450, −22.24805606149515502792494690844, −21.41822943322396758650423221622, −20.32207193066252761740563224828, −19.44463561430619581486148763265, −19.048020690657769954611867185943, −17.81611456910409732564877668585, −17.49564375037653120911506744355, −16.21882387577317267308024915454, −15.46422219256460541637338882454, −14.31702403103929781051279798694, −13.448268890971837853433254051320, −11.96489076877496058337474147540, −11.66730825789346243328512636426, −10.56336002897283208792767118103, −9.79037642553492400899637028738, −8.91425161989566257730529707491, −7.62469595781329805531727826794, −7.14117598929869053913781254764, −5.932680516634665478297589402258, −4.274855801944762828468894852837, −3.145677113144361060080349929917, −2.44523917040251896419864215558, −0.81332968671421835074987283098, 1.06285790282199492272017649919, 2.07026893878881387249922029820, 3.99643779731975713405731540512, 4.98233099708697555887816916360, 6.053637303838026730063975464, 7.0591287461109681298820733642, 8.067143548890563873519772103143, 8.83056660390952910478323557998, 9.6882655084278459659882196995, 10.507279219312227253349389805371, 11.99111740445398162656557441226, 12.32372080230205444366189716405, 13.99222501829436276380020668372, 14.60002896653433994877706907878, 15.70742044542336669036023993226, 16.498974407207087856944214054197, 17.11885042013855938405312584214, 17.865243530698717960696662818455, 19.13862404524254965493567513815, 19.64222749086849024868439774133, 20.45028697710090110880031085927, 21.41486635316669681797392948434, 22.67092583258916913385174126304, 23.5506594090285986989484445728, 24.442942726788054674289067808970

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