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Properties

Label 1-21e2-441.191-r1-0-0
Degree $1$
Conductor $441$
Sign $0.700 - 0.713i$
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  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)5-s + (−0.623 + 0.781i)8-s + (−0.988 + 0.149i)10-s + (0.222 − 0.974i)11-s + (−0.733 − 0.680i)13-s + (−0.988 + 0.149i)16-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.0747 − 0.997i)26-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)5-s + (−0.623 + 0.781i)8-s + (−0.988 + 0.149i)10-s + (0.222 − 0.974i)11-s + (−0.733 − 0.680i)13-s + (−0.988 + 0.149i)16-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.0747 − 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.700 - 0.713i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9000135190 - 0.3775012306i\)
\(L(\frac12)\) \(\approx\) \(0.9000135190 - 0.3775012306i\)
\(L(1)\) \(\approx\) \(1.032194430 + 0.4599641075i\)
\(L(1)\) \(\approx\) \(1.032194430 + 0.4599641075i\)

Euler product

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

−23.67756279318117881046013788974, −23.12333965681254327089418946094, −22.30232311259381355632665609143, −21.26320264908947005280146967537, −20.516875990278354478608051158, −19.887633200743054089581227473652, −19.08774862660008346107143682754, −18.14560465208870561851519966224, −16.82834116333640458863782964910, −16.085929228562753915052616930150, −14.92677211205618144113767690909, −14.43853429452843850358082089487, −13.12622422704885139594588582974, −12.45895234229568303994863215715, −11.808772257167724742938179087709, −10.88231639773702055550159413225, −9.64666678241797351203284894958, −9.0479070101329722115259326028, −7.56253228468914264341615477343, −6.632886690638176395700697258030, −5.16389568457090340369296039307, −4.6082858832165975923139158246, −3.64996943826098123778501351851, −2.31199144520926720260348622399, −1.21795848730741028534465459951, 0.20465308205764386620006210571, 2.46971247788699383550507350600, 3.398177560419605797907462945834, 4.265685330712423764443974002858, 5.5316409596094097442798612672, 6.397508320409849219774360237768, 7.37621778344702172961477294283, 8.09266114874203853921369081553, 9.17220165295960931173026514407, 10.75481565842386577413175991592, 11.35471205120333848310496411316, 12.4441196960372956049017268954, 13.299517368401162172107309180414, 14.247872319066419084021228271304, 15.22379157525077711661835856507, 15.43422827719517925296049626944, 16.85399148145870921966779745471, 17.32135023512599986960043873377, 18.612824544874491240429208086407, 19.39014848632873886794016701768, 20.3882326600407389166730632561, 21.6329198812513957532038179219, 22.08645716000516973131559302580, 22.87811060099727498323573222710, 23.87206680023095840320376372998

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