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Properties

Label 1-1081-1081.105-r1-0-0
Degree $1$
Conductor $1081$
Sign $0.805 - 0.592i$
Analytic cond. $116.169$
Root an. cond. $116.169$
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  + (0.997 + 0.0744i)2-s + (0.154 + 0.987i)3-s + (0.988 + 0.148i)4-s + (0.239 − 0.970i)5-s + (0.0806 + 0.996i)6-s + (0.969 − 0.245i)7-s + (0.975 + 0.221i)8-s + (−0.952 + 0.305i)9-s + (0.311 − 0.950i)10-s + (0.263 − 0.964i)11-s + (0.00620 + 0.999i)12-s + (−0.867 − 0.498i)13-s + (0.984 − 0.172i)14-s + (0.996 + 0.0868i)15-s + (0.955 + 0.293i)16-s + (0.767 − 0.640i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0744i)2-s + (0.154 + 0.987i)3-s + (0.988 + 0.148i)4-s + (0.239 − 0.970i)5-s + (0.0806 + 0.996i)6-s + (0.969 − 0.245i)7-s + (0.975 + 0.221i)8-s + (−0.952 + 0.305i)9-s + (0.311 − 0.950i)10-s + (0.263 − 0.964i)11-s + (0.00620 + 0.999i)12-s + (−0.867 − 0.498i)13-s + (0.984 − 0.172i)14-s + (0.996 + 0.0868i)15-s + (0.955 + 0.293i)16-s + (0.767 − 0.640i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1081\)    =    \(23 \cdot 47\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(116.169\)
Root analytic conductor: \(116.169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1081} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1081,\ (1:\ ),\ 0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.025399334 - 1.647487518i\)
\(L(\frac12)\) \(\approx\) \(5.025399334 - 1.647487518i\)
\(L(1)\) \(\approx\) \(2.373822720 + 0.0007279187582i\)
\(L(1)\) \(\approx\) \(2.373822720 + 0.0007279187582i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.997 + 0.0744i)T \)
3 \( 1 + (0.154 + 0.987i)T \)
5 \( 1 + (0.239 - 0.970i)T \)
7 \( 1 + (0.969 - 0.245i)T \)
11 \( 1 + (0.263 - 0.964i)T \)
13 \( 1 + (-0.867 - 0.498i)T \)
17 \( 1 + (0.767 - 0.640i)T \)
19 \( 1 + (-0.783 + 0.621i)T \)
29 \( 1 + (0.191 - 0.981i)T \)
31 \( 1 + (-0.154 + 0.987i)T \)
37 \( 1 + (0.0310 - 0.999i)T \)
41 \( 1 + (0.873 - 0.487i)T \)
43 \( 1 + (0.535 + 0.844i)T \)
53 \( 1 + (0.700 - 0.713i)T \)
59 \( 1 + (-0.907 + 0.421i)T \)
61 \( 1 + (-0.896 + 0.443i)T \)
67 \( 1 + (0.820 - 0.571i)T \)
71 \( 1 + (-0.977 - 0.209i)T \)
73 \( 1 + (0.966 - 0.257i)T \)
79 \( 1 + (-0.791 - 0.611i)T \)
83 \( 1 + (0.901 - 0.432i)T \)
89 \( 1 + (0.503 - 0.863i)T \)
97 \( 1 + (-0.426 - 0.904i)T \)
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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

−21.57089985958093022839211665077, −20.60051372391402033557798309392, −19.82121517003161367194906000414, −19.06878971102779093373484218905, −18.381689612630868107818112358555, −17.363983097659104569256169568188, −16.96675726313088000593346496437, −15.29782113983189684689483213685, −14.76760024785497185106395393402, −14.37599256419846922235756303286, −13.58397769677379181959538097433, −12.57096918507966133162193383764, −12.05660975271161971897126769246, −11.25780187278281626194261802269, −10.51928647171281725814387164299, −9.38508552779562279039288206607, −8.026020424152882189518523150010, −7.339302940325862208220061733900, −6.72142300154340653980913800902, −5.89577745717775084381374676023, −4.94098011592127854893977593324, −3.95141526596322333987345794687, −2.69218276757776013245011437830, −2.158379670605726218105578808336, −1.36335481606144902942058513336, 0.67492578299809526101908022630, 1.96221616385512904260690403443, 3.01468093757093720546003145894, 4.048669534118547040833289528577, 4.670538127571382806880921647376, 5.43197200756141450161212987533, 5.97983544503747599591708708634, 7.59065439681228619176049495195, 8.20129061458378220629590730193, 9.13657927691215782555832765435, 10.20288109834697018675108438096, 10.90041952971255771005240153936, 11.79729387785867638261451427008, 12.428353950503390718064515072526, 13.5424694723782636542357217688, 14.25067085699671773253127082059, 14.69297004334368509598656256049, 15.70449821797204192772001661816, 16.43618144206747669311240801417, 16.93162072486244795323130745400, 17.68452376595945855373455490148, 19.332970604863202635419362843969, 19.928185801471733688715465335308, 20.7803200472288166062653599059, 21.28153015179546821217091310193

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