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Properties

Label 1-1287-1287.1226-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.498 + 0.867i$
Analytic cond. $138.307$
Root an. cond. $138.307$
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  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.913 − 0.406i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (0.978 + 0.207i)19-s + (−0.809 + 0.587i)20-s − 23-s + (0.669 − 0.743i)25-s + (0.104 − 0.994i)28-s + (−0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.913 − 0.406i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (0.978 + 0.207i)19-s + (−0.809 + 0.587i)20-s − 23-s + (0.669 − 0.743i)25-s + (0.104 − 0.994i)28-s + (−0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.498 + 0.867i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.498 + 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9639268020 + 0.5577904880i\)
\(L(\frac12)\) \(\approx\) \(0.9639268020 + 0.5577904880i\)
\(L(1)\) \(\approx\) \(0.9041725126 - 0.2805905612i\)
\(L(1)\) \(\approx\) \(0.9041725126 - 0.2805905612i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.978 - 0.207i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−20.6033103596114043100303903680, −19.94698980782824601455854712428, −18.91016371410902279125948793632, −18.11269351682045283885601308342, −17.64379023213661721292668380848, −16.84003587433669654868224907090, −16.21496483965937893780920920287, −15.374030923926515787536599482896, −14.47921348259750939710268934207, −13.68849541030766093906218243917, −13.49763264132479389044450602787, −12.44896942745646905941644731229, −11.07077667824418350281679041155, −10.361225684392714922518374490228, −9.496221738203568593212266878739, −9.071207024134033472681706287427, −7.66332459691621425161013199840, −7.25232683707407338550107848501, −6.29710293253579562034912485083, −5.72979436361322074118162325376, −4.65166040745099402979924265482, −3.8351953073976999509657991403, −2.66584830463813725578523855750, −1.33637197168293279883387882315, −0.245283874477425356791199722942, 1.06493308065579487453886667605, 2.07092551400037926012089335142, 2.6321587922370459280809325570, 3.776464739290697963449596524868, 4.78639338431985380879349166268, 5.63101906812248654567752409644, 6.28937115819254162800924137049, 7.7796126801187262187987982114, 8.68804694672790453439056804181, 9.32928699441403626752497763386, 9.88286671207382147461953276862, 10.786029753262826479998183144521, 11.73273028034621300482289232302, 12.38938671992005529594381131592, 13.10600398544272421242776647930, 13.76117092414196940325840901729, 14.56613208157875560797954466431, 15.622602701667751490073868954991, 16.5210551419383333500453520141, 17.38188472914832872021509246418, 18.19227289291783713908285876755, 18.44556268265482564032081890930, 19.696284008457048482790820717526, 20.05050746089864065843466790706, 21.107335710815528681500604831274

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