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Properties

Label 1-21e2-441.158-r1-0-0
Degree $1$
Conductor $441$
Sign $0.569 + 0.822i$
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  + (−0.826 − 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

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

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.569 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8010193166 + 0.4197077989i\)
\(L(\frac12)\) \(\approx\) \(0.8010193166 + 0.4197077989i\)
\(L(1)\) \(\approx\) \(0.7320757977 - 0.1318716810i\)
\(L(1)\) \(\approx\) \(0.7320757977 - 0.1318716810i\)

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

−23.87091066276542973674214351272, −22.76990834138080307087305719333, −22.36928005429316671112791166701, −20.98214623322358679751953265765, −20.02337765322481717805471302307, −19.28730210468366970017654033140, −18.253234700338676447220410284747, −17.85218825322210581884760409058, −16.92452399941667705701517555997, −15.82673018011329601443727773416, −15.15385821905493088158624413153, −14.280317133068038448296431632675, −13.52104193960688884337478701480, −11.945219985469376422466141513379, −11.00156791786905531584640126805, −10.31060231584060092707160055898, −9.324677270733364580991978490950, −8.47919112627890909821527363922, −7.26154840152647370615171735055, −6.63401432392856516261955461243, −5.7936383350856448673369040457, −4.41088339011434588387736176013, −2.90013843572889682598214578283, −1.793773154545198151421864088233, −0.33006567431137937163511249610, 1.230669030894256114338247948158, 1.81619276082896097062922724562, 3.54420532268699508841826120674, 4.26781311102234673165136044212, 5.86578570429871424621489040902, 6.79020045855726749101016453316, 8.3282619715340908460311327395, 8.621027301676920779348929163006, 9.6231270143363776256573917546, 10.56097860787005094735433277626, 11.6242649918860212740541206657, 12.30419106409639494632426295482, 13.24230435724803667784541619943, 14.15614085664263102045088058133, 15.673757188440918847403331889462, 16.413545165844927996831248024275, 17.13137801393893508262710176256, 17.85095749724363192355950636620, 18.98900077885883420352562975053, 19.65276323300471396734353977082, 20.41873960065842011879871887814, 21.4191232467000730354950296038, 21.729420866183330653185802256531, 23.16936444602014520524546037265, 24.16097416605368664698333351124

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