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Properties

Label 1-2032-2032.781-r0-0-0
Degree $1$
Conductor $2032$
Sign $0.880 - 0.474i$
Analytic cond. $9.43656$
Root an. cond. $9.43656$
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  + (0.866 − 0.5i)3-s − i·5-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + i·19-s + (0.866 + 0.5i)21-s + (0.5 + 0.866i)23-s − 25-s − i·27-s + (−0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s − i·5-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + i·19-s + (0.866 + 0.5i)21-s + (0.5 + 0.866i)23-s − 25-s − i·27-s + (−0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2032\)    =    \(2^{4} \cdot 127\)
Sign: $0.880 - 0.474i$
Analytic conductor: \(9.43656\)
Root analytic conductor: \(9.43656\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2032} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2032,\ (0:\ ),\ 0.880 - 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.683386271 - 0.6776966777i\)
\(L(\frac12)\) \(\approx\) \(2.683386271 - 0.6776966777i\)
\(L(1)\) \(\approx\) \(1.631240806 - 0.3159659518i\)
\(L(1)\) \(\approx\) \(1.631240806 - 0.3159659518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
127 \( 1 \)
good3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 - iT \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + iT \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 - T \)
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

−19.98513028645734051183478373332, −19.374151646676930655173828084702, −18.555881400120095236580928252070, −17.97459219752257919625683271300, −16.960393884148463028634967851118, −16.24845158694484264244969779328, −15.55206107317948326195363292284, −14.59574616565908297736796226754, −14.25795452878307431092052319581, −13.67318326343151115079421951623, −12.92688503521711384996581296106, −11.3447836424735987320874788754, −11.14258235841702938169712502437, −10.45698254495342675524239987049, −9.38540538549742561200298466185, −8.915311480266135863411412954806, −8.00453363377046459810492565486, −6.99414623975581496348455599524, −6.78228790440217506739864328343, −5.41383231198054341389026377219, −4.29929411655262070605139918601, −3.856733463435400060668250494411, −2.96836901108421842502979620639, −2.156457758591156620946869362485, −1.01638011159607885646224110075, 1.15050945785760901716817536468, 1.68751219824408529342957600021, 2.544135274947005156632549117039, 3.904021387981839490429962394296, 4.17073468672435792425138338385, 5.68127662061326325379360705589, 5.959615943427338896373329929579, 7.29465271575136374093094976226, 7.99619161923663497977846385810, 8.69150929160286385526632835733, 9.14471164170978611632251099819, 9.897236380402951067965657842774, 11.20178726474909838377658406268, 11.92183889424265653797402007626, 12.66835741914378903815551944147, 13.12006204913940532121756262030, 13.955159473822753712923225365063, 14.93038342899733417620247801075, 15.21662987002910587323628248693, 16.117154334650957724760458262891, 17.16079726144547134745271122556, 17.6528072422109298007228584710, 18.54456233103623774316212407527, 19.1423965630987601753598055530, 19.95050354756562899553535627489

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