Properties

Label 1-448-448.405-r1-0-0
Degree $1$
Conductor $448$
Sign $0.634 + 0.773i$
Analytic cond. $48.1442$
Root an. cond. $48.1442$
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  + (0.923 − 0.382i)3-s + (−0.382 + 0.923i)5-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.382 − 0.923i)13-s + i·15-s i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + (0.923 − 0.382i)29-s + 31-s + 33-s + (−0.382 + 0.923i)37-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.382 + 0.923i)5-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.382 − 0.923i)13-s + i·15-s i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + (0.923 − 0.382i)29-s + 31-s + 33-s + (−0.382 + 0.923i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.634 + 0.773i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ 0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.460237222 + 1.163605546i\)
\(L(\frac12)\) \(\approx\) \(2.460237222 + 1.163605546i\)
\(L(1)\) \(\approx\) \(1.465554581 + 0.1916271039i\)
\(L(1)\) \(\approx\) \(1.465554581 + 0.1916271039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.923 + 0.382i)T \)
59 \( 1 + (-0.382 + 0.923i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.382 + 0.923i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + T \)
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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

−23.97041173022086527427544885350, −22.75018541536192159265283785269, −21.71252707883428382269467179095, −21.119314204139453239319476476191, −20.00559929097875308906870220202, −19.73850690631916205510685288826, −18.78471634210296718830170857461, −17.55915725547723421498629350857, −16.30035123576500212242395480650, −16.145732836194491463392800650732, −14.89990682589818117352605502175, −14.035652001954074424836596279948, −13.38640131796319552624738516508, −12.14392805467011735863600267521, −11.48730960346634326454035043851, −10.056719958306537302747350995998, −9.13998945537040479077306781671, −8.68984844928777632471326815289, −7.59592275392707582457432785554, −6.5797949448848740066309413813, −4.92942291038651190037501566417, −4.36064082321187898289048363726, −3.2580262827992762114360968198, −2.01988727010445815988628167976, −0.69788098751995420033116329901, 1.21634671098759452439246462943, 2.42457217225829214429236628658, 3.43124904342540537948355395634, 4.18556645314043620689514358578, 5.95382422631339436261070885269, 6.85622338819227692787688537943, 7.776799314003492742213434678931, 8.42972533946280828876001694874, 9.80827470624800255505598533791, 10.334540068089939522815552269302, 11.806235863997117261113799529703, 12.394044734835477676904110678776, 13.63374245881936680617524867158, 14.367826857366634673799958273087, 15.0890682118223666794074169795, 15.73508915942301075415405935720, 17.27216883176757094339049818329, 17.96451337977873223844678975950, 18.98703387358744458167681714384, 19.56682303355015292673022045903, 20.24105996432646963175614674991, 21.33634613369611641734429903273, 22.27803075500873152762277896198, 23.02142476268424052610175220395, 24.00209856665887500192043450106

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