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Properties

Label 1-21e2-441.365-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.328 + 0.944i$
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  + (−0.365 − 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

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

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.328 + 0.944i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3579820136 - 0.5035776658i\)
\(L(\frac12)\) \(\approx\) \(-0.3579820136 - 0.5035776658i\)
\(L(1)\) \(\approx\) \(0.5074529886 - 0.5020715225i\)
\(L(1)\) \(\approx\) \(0.5074529886 - 0.5020715225i\)

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

−24.576185732610478568335453381587, −23.51269352526360806486836951604, −22.94948801079728613622283363813, −22.13787661749473535696239346035, −21.21985742445773929138778788156, −19.614181688340053852560831080632, −19.34631813027014920272786138816, −17.98146605828348131922974606035, −17.78782743259327108878776997099, −16.65796345217073615983077110045, −15.62457958690322157793028796836, −14.89928563822080287908584279397, −14.342126235218082720711080950779, −13.265410411019607557835614771143, −12.15390620836847748865256646853, −10.867735183086758271800370949411, −10.016983265647332765458712371716, −9.346076392768156216580486819532, −7.885962225291770020094913317302, −7.350772428290483853640267672886, −6.480725179235643954185236796202, −5.39195211897056296856923667180, −4.38070946265358244537015060431, −3.007518690846159992976535060690, −1.59099183117075237866958872475, 0.21732902714057139541059236955, 1.060309044898543275880252863348, 2.4903617149776106538376542185, 3.46319212692639164046762620, 4.80068009653125524180929441917, 5.37205286980457535833840540326, 7.22894888694481373502889852532, 8.16767368254981107472938288856, 9.02417393394307355455186392703, 9.774554827530635804380314872214, 10.821026986919586873191373432, 11.84978671111112134373321420783, 12.41903423002839230804050818532, 13.43661926272481634788189275139, 14.09973949100380033147954758804, 15.62834221944664224783137185724, 16.61554052300426877214552312983, 17.1183255922140539502328724762, 18.3112740989088240051640562380, 18.95376085865737635548451124524, 20.06416988533883075622430484050, 20.45374354087490875582211054214, 21.41128453348537947515035163510, 22.15015985896478258625904060565, 23.10092506408857994590872279291

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