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Properties

Label 1-6017-6017.1040-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.507 + 0.861i$
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  + (0.0563 − 0.998i)2-s + (−0.309 − 0.951i)3-s + (−0.993 − 0.112i)4-s + (−0.619 + 0.784i)5-s + (−0.966 + 0.254i)6-s + (0.152 − 0.988i)7-s + (−0.168 + 0.985i)8-s + (−0.809 + 0.587i)9-s + (0.748 + 0.663i)10-s + (0.200 + 0.979i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.937 + 0.347i)15-s + (0.974 + 0.223i)16-s + (0.339 − 0.940i)17-s + (0.541 + 0.840i)18-s + ⋯
L(s)  = 1  + (0.0563 − 0.998i)2-s + (−0.309 − 0.951i)3-s + (−0.993 − 0.112i)4-s + (−0.619 + 0.784i)5-s + (−0.966 + 0.254i)6-s + (0.152 − 0.988i)7-s + (−0.168 + 0.985i)8-s + (−0.809 + 0.587i)9-s + (0.748 + 0.663i)10-s + (0.200 + 0.979i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.937 + 0.347i)15-s + (0.974 + 0.223i)16-s + (0.339 − 0.940i)17-s + (0.541 + 0.840i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.507 + 0.861i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1040, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3683653055 - 0.6447193118i\)
\(L(\frac12)\) \(\approx\) \(-0.3683653055 - 0.6447193118i\)
\(L(1)\) \(\approx\) \(0.4370206718 - 0.5583651187i\)
\(L(1)\) \(\approx\) \(0.4370206718 - 0.5583651187i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.0563 - 0.998i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.619 + 0.784i)T \)
7 \( 1 + (0.152 - 0.988i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (0.339 - 0.940i)T \)
19 \( 1 + (-0.0884 + 0.996i)T \)
23 \( 1 + (-0.428 - 0.903i)T \)
29 \( 1 + (-0.958 + 0.285i)T \)
31 \( 1 + (-0.485 - 0.873i)T \)
37 \( 1 + (0.726 + 0.686i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.948 + 0.316i)T \)
47 \( 1 + (-0.384 - 0.923i)T \)
53 \( 1 + (0.997 - 0.0643i)T \)
59 \( 1 + (-0.0884 - 0.996i)T \)
61 \( 1 + (0.966 - 0.254i)T \)
67 \( 1 + (0.948 + 0.316i)T \)
71 \( 1 + (0.999 - 0.0161i)T \)
73 \( 1 + (0.962 + 0.270i)T \)
79 \( 1 + (0.215 - 0.976i)T \)
83 \( 1 + (0.369 + 0.929i)T \)
89 \( 1 + (0.354 - 0.935i)T \)
97 \( 1 + (-0.136 + 0.990i)T \)
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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

−17.7163588070656833202852775379, −17.40723896122382134733281586506, −16.61356209736637476884620642882, −16.17398772628292517096093282421, −15.57214567375209405585116104408, −14.978234744160200421752810689743, −14.68039351318517478494680847772, −13.74173082060795125314657235722, −12.68013953110186357487973696001, −12.43307366161515685984443919581, −11.564829525792540497462966956493, −10.971891599163147787358535504498, −9.81081505832970668099961253971, −9.30829988309348558126778675233, −8.91519482578036569334843422353, −8.12660681777125327890898830990, −7.54162424064866169868545439480, −6.592259503698613183511645970544, −5.601800332130916639811706825979, −5.445450537300306840842197215393, −4.594250785710888904349180220802, −4.085832990991595943018312604400, −3.34511366130809684892509351254, −2.234188274218643438014127377039, −0.89902996524056923937710490614, 0.306376572587560125125591116275, 0.846765617123799242877175239839, 2.06363137150579017036186459983, 2.49144355663038505220763077666, 3.44671239616995341397079986020, 4.01308966535756051125169169572, 4.920813364732524637557200070260, 5.648065360096850214153934717318, 6.57478644813366667317681896804, 7.370034864117656080810167277107, 7.79332220718739189971242049370, 8.36981367267351514864116377766, 9.61364026647757035027733916990, 10.120485722574997853995301462189, 10.92465869792810219023238868860, 11.25997011884065543495416532009, 12.037139477967739684408102935571, 12.51054689680344639405271416134, 13.19100531039682942919648719929, 13.96427685828217801304697864595, 14.45537268128718856494068449629, 14.869276696370914208099968581166, 16.22448611546594054340362614494, 16.84692858148486944749868627781, 17.438679658109533502388084149788

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