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Properties

Label 1-21e2-441.328-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.908 - 0.417i$
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 + (0.733 − 0.680i)5-s + (−0.222 − 0.974i)8-s + (0.222 − 0.974i)10-s + (0.826 − 0.563i)11-s + (−0.0747 − 0.997i)13-s + (−0.733 − 0.680i)16-s + (−0.623 + 0.781i)17-s − 19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.623 − 0.781i)26-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (0.733 − 0.680i)5-s + (−0.222 − 0.974i)8-s + (0.222 − 0.974i)10-s + (0.826 − 0.563i)11-s + (−0.0747 − 0.997i)13-s + (−0.733 − 0.680i)16-s + (−0.623 + 0.781i)17-s − 19-s + (−0.365 − 0.930i)20-s + (0.365 − 0.930i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.623 − 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.908 - 0.417i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7471064188 - 3.413103134i\)
\(L(\frac12)\) \(\approx\) \(0.7471064188 - 3.413103134i\)
\(L(1)\) \(\approx\) \(1.441116181 - 1.216548278i\)
\(L(1)\) \(\approx\) \(1.441116181 - 1.216548278i\)

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 \( 1 + (0.733 - 0.680i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (-0.0747 - 0.997i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
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 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.0747 + 0.997i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
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

−24.31026671180686482264848542041, −23.12343423515366987780659651854, −22.6749211511122361464001211271, −21.62136558953353850200910111131, −21.27031609302186683761491283738, −20.08346760188696365646682839482, −19.041388678695830907535265222186, −17.878633619010542823361298672367, −17.211298281313526142435860481619, −16.42297905982520427926271739811, −15.15780131292710118022138015121, −14.70245421685331107404603981577, −13.691098022080556652735421072789, −13.17998786966288544263240334340, −11.79767495075625116182519472785, −11.29583108495379140826556973392, −9.844181779591895594585555843392, −9.013162603108489784373873376505, −7.665513396976438042026749620029, −6.622494394671559376379572423108, −6.28068774577989310882839033383, −4.87798160462652088701179867620, −4.05015464037117064087292447538, −2.74936530506440429076668900826, −1.83048526452485831710567898802, 0.65038992797372546277801189029, 1.71383593642110199714795679221, 2.83886911257685802703289320714, 4.06016826700425122646774067947, 4.97624704390353488566414364664, 5.98541180412533751911734069363, 6.65292136695206159898737166347, 8.4216254993152079683679480366, 9.202433057229282380704300309105, 10.40921578383606802899614114772, 10.94980828910603869805382654071, 12.41113910910539330439389558076, 12.71894036532678714699276650108, 13.74617993777191031062257827617, 14.508655369607868933012164030, 15.43181924474558605261254866953, 16.5208597701276943648530355383, 17.36886918302904577658940013261, 18.39645838384314984028878771606, 19.62551518492293516795996376277, 20.01545907625792560572472234249, 21.141272176186894712771718046886, 21.64879433596085830594445461764, 22.46958723676474666056074814485, 23.39081461120185647099524293098

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