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Properties

Label 1-6017-6017.1076-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.753 + 0.657i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (−0.463 − 0.885i)2-s + (0.134 + 0.990i)3-s + (−0.569 + 0.821i)4-s + (−0.996 − 0.0850i)5-s + (0.815 − 0.578i)6-s + (−0.0264 − 0.999i)7-s + (0.992 + 0.123i)8-s + (−0.963 + 0.266i)9-s + (0.386 + 0.922i)10-s + (−0.890 − 0.454i)12-s + (−0.946 − 0.323i)13-s + (−0.873 + 0.486i)14-s + (−0.0494 − 0.998i)15-s + (−0.350 − 0.936i)16-s + (−0.972 + 0.234i)17-s + (0.682 + 0.730i)18-s + ⋯
L(s)  = 1  + (−0.463 − 0.885i)2-s + (0.134 + 0.990i)3-s + (−0.569 + 0.821i)4-s + (−0.996 − 0.0850i)5-s + (0.815 − 0.578i)6-s + (−0.0264 − 0.999i)7-s + (0.992 + 0.123i)8-s + (−0.963 + 0.266i)9-s + (0.386 + 0.922i)10-s + (−0.890 − 0.454i)12-s + (−0.946 − 0.323i)13-s + (−0.873 + 0.486i)14-s + (−0.0494 − 0.998i)15-s + (−0.350 − 0.936i)16-s + (−0.972 + 0.234i)17-s + (0.682 + 0.730i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1076, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.753 + 0.657i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03597695090 + 0.01348262689i\)
\(L(\frac12)\) \(\approx\) \(0.03597695090 + 0.01348262689i\)
\(L(1)\) \(\approx\) \(0.4339486303 - 0.1332037290i\)
\(L(1)\) \(\approx\) \(0.4339486303 - 0.1332037290i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.463 - 0.885i)T \)
3 \( 1 + (0.134 + 0.990i)T \)
5 \( 1 + (-0.996 - 0.0850i)T \)
7 \( 1 + (-0.0264 - 0.999i)T \)
13 \( 1 + (-0.946 - 0.323i)T \)
17 \( 1 + (-0.972 + 0.234i)T \)
19 \( 1 + (-0.588 - 0.808i)T \)
23 \( 1 + (-0.976 + 0.216i)T \)
29 \( 1 + (-0.999 + 0.0207i)T \)
31 \( 1 + (0.295 - 0.955i)T \)
37 \( 1 + (0.718 - 0.695i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (-0.717 - 0.696i)T \)
47 \( 1 + (-0.231 + 0.972i)T \)
53 \( 1 + (-0.988 + 0.153i)T \)
59 \( 1 + (-0.657 - 0.753i)T \)
61 \( 1 + (-0.745 - 0.666i)T \)
67 \( 1 + (0.838 - 0.544i)T \)
71 \( 1 + (0.564 + 0.825i)T \)
73 \( 1 + (-0.547 - 0.837i)T \)
79 \( 1 + (-0.938 - 0.344i)T \)
83 \( 1 + (0.997 - 0.0643i)T \)
89 \( 1 + (-0.952 + 0.305i)T \)
97 \( 1 + (0.0471 - 0.998i)T \)
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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

−17.754434880136896543660952885182, −16.92159923953556293355909821977, −16.41940245691263533456362705086, −15.61120194007484403448827068038, −14.9643090740273942579801116681, −14.64842966268130066616182345738, −13.8515931819803710912375990189, −13.03781035083773512557424290951, −12.36631609076434660652548452800, −11.80229885963010823571990440764, −11.17805800377031771128111119932, −10.19801896506931375222174336702, −9.35712586005955067648489769492, −8.65748773193173679629034645302, −8.185148421844035336653269519595, −7.67068956818838668118862416358, −6.78352877513609331746521815425, −6.502148855993597800647020455042, −5.59942578276258222716098177160, −4.86867603798415948850441921448, −4.07621901035044950860442467264, −2.9976434917993434352381266413, −2.13172455238473216200834575629, −1.454965948587019214588983682015, −0.030662443660309242393033138715, 0.334137457856368868005239978491, 1.78783547688841663253913145215, 2.643784378582975041822191335264, 3.412726148601554052193049380877, 4.054103578635761943532703959272, 4.49299494674160205623514173886, 5.07367567145122072789791284095, 6.416802626196026454456046916106, 7.3861151814799089786710131136, 7.91596518752697305527302738710, 8.49411237745729176235993899158, 9.39806011171652734540550412949, 9.77671769880198326474571449051, 10.65741692619890907239440446296, 11.05938211791620220839301445577, 11.53141475645550537153114372725, 12.40556898624501164716441888483, 13.08573503148099837170853775005, 13.769874581510702712590928806352, 14.5782250133277060699902703381, 15.26609296798964138833036151917, 15.86776192954014523092680207011, 16.65242384472429137030941419337, 17.137707649471368568381851941143, 17.58503680751060895333020799751

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