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Properties

Label 1-21e2-441.200-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  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.997i)5-s + (0.900 + 0.433i)8-s + (0.826 + 0.563i)10-s + (−0.365 − 0.930i)11-s + (0.365 + 0.930i)13-s + (−0.900 + 0.433i)16-s + (−0.955 + 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.955 + 0.294i)20-s + (0.955 + 0.294i)22-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.955 − 0.294i)26-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.997i)5-s + (0.900 + 0.433i)8-s + (0.826 + 0.563i)10-s + (−0.365 − 0.930i)11-s + (0.365 + 0.930i)13-s + (−0.900 + 0.433i)16-s + (−0.955 + 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.955 + 0.294i)20-s + (0.955 + 0.294i)22-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.955 − 0.294i)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: $\chi_{441} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.0142 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5895055260 + 0.5811657651i\)
\(L(\frac12)\) \(\approx\) \(0.5895055260 + 0.5811657651i\)
\(L(1)\) \(\approx\) \(0.6771666628 + 0.1411118233i\)
\(L(1)\) \(\approx\) \(0.6771666628 + 0.1411118233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (-0.955 + 0.294i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.365 + 0.930i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
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

−23.267743387116713058684524009738, −22.62285726426612529685020826350, −21.92154504051614581068845779325, −20.90335846269848583763073006504, −20.148402772640798845805565046065, −19.301457619410558695949145287198, −18.48746657626581491157107727763, −17.69908543437015018695730680732, −17.19814410557949726370013442035, −15.55335076137800274750987306097, −15.23633035464787137110703354723, −13.66525210170002019484758546860, −13.05538627654784671017940375858, −11.91177813261140883263838710516, −11.02497616873118594046447416953, −10.40714439852841876680578743748, −9.50686867235821710106894438670, −8.432545021758066152687758722321, −7.39252752561172459487800151954, −6.707469293724202184417012447865, −5.07239435045931810112539704521, −3.8444721571387209681835120038, −2.82174184673726638833244595397, −1.99610354235376071163015967532, −0.34767370432199138658500332424, 0.87468451766151817583084204756, 2.01951043432122109737565832975, 3.95907788187711210895048602353, 4.93408546818113386754078716713, 5.93367064425035739725202244365, 6.79261823743452887710138213868, 8.13321751181919896697022083306, 8.64520659364267525518012603763, 9.43683521824955079710298519751, 10.63693830245101308254660202060, 11.478665645869523096472498952, 12.84562054532215784107076958505, 13.61361458117337597558980634086, 14.578054021121660197091599184183, 15.64621301522617903911366461530, 16.399578953293418215077674601712, 16.88855506665720149191248198468, 17.89703804524076242088454769434, 18.93460557566268156771328077185, 19.44442005176807072801499548596, 20.6263038864160084396758586472, 21.295281504818972010551956356694, 22.6093575187962269438097576769, 23.53999663365505290876867631971, 24.2857482471770283810282658410

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