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Properties

Label 1-6017-6017.350-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.00267 - 0.999i$
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  + (−0.943 + 0.331i)2-s + (0.309 − 0.951i)3-s + (0.779 − 0.626i)4-s + (0.644 − 0.764i)5-s + (0.0241 + 0.999i)6-s + (−0.607 + 0.794i)7-s + (−0.527 + 0.849i)8-s + (−0.809 − 0.587i)9-s + (−0.354 + 0.935i)10-s + (−0.354 − 0.935i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.527 − 0.849i)15-s + (0.215 − 0.976i)16-s + (0.485 + 0.873i)17-s + (0.958 + 0.285i)18-s + ⋯
L(s)  = 1  + (−0.943 + 0.331i)2-s + (0.309 − 0.951i)3-s + (0.779 − 0.626i)4-s + (0.644 − 0.764i)5-s + (0.0241 + 0.999i)6-s + (−0.607 + 0.794i)7-s + (−0.527 + 0.849i)8-s + (−0.809 − 0.587i)9-s + (−0.354 + 0.935i)10-s + (−0.354 − 0.935i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.527 − 0.849i)15-s + (0.215 − 0.976i)16-s + (0.485 + 0.873i)17-s + (0.958 + 0.285i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00267 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00267 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.00267 - 0.999i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.00267 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7571146857 - 0.7591461693i\)
\(L(\frac12)\) \(\approx\) \(0.7571146857 - 0.7591461693i\)
\(L(1)\) \(\approx\) \(0.7328067199 - 0.2281520291i\)
\(L(1)\) \(\approx\) \(0.7328067199 - 0.2281520291i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.943 + 0.331i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.644 - 0.764i)T \)
7 \( 1 + (-0.607 + 0.794i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.485 + 0.873i)T \)
19 \( 1 + (-0.861 + 0.506i)T \)
23 \( 1 + (0.885 - 0.464i)T \)
29 \( 1 + (-0.168 + 0.985i)T \)
31 \( 1 + (0.995 - 0.0965i)T \)
37 \( 1 + (-0.168 + 0.985i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.354 - 0.935i)T \)
47 \( 1 + (0.715 - 0.698i)T \)
53 \( 1 + (0.926 + 0.377i)T \)
59 \( 1 + (-0.861 - 0.506i)T \)
61 \( 1 + (0.0241 + 0.999i)T \)
67 \( 1 + (-0.354 - 0.935i)T \)
71 \( 1 + (0.995 + 0.0965i)T \)
73 \( 1 + (-0.0724 - 0.997i)T \)
79 \( 1 + (-0.262 + 0.964i)T \)
83 \( 1 + (0.644 - 0.764i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (-0.681 + 0.732i)T \)
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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

−17.687615679523741708104517090949, −17.1588159601018981049428441501, −16.76669294797086405184745605943, −16.054308093178920159427290060615, −15.352944722924796983516839565108, −14.73005405444669863552526756799, −13.98940669550204548943462934750, −13.388373597854089283984490016449, −12.569253154393489287869612427963, −11.45167134048222655249650304467, −11.162613403481741438400767543063, −10.33171426763027575273890441280, −9.87665270085206184929826066214, −9.4584611626745778211152208148, −8.87401823295725955926689527053, −7.79860662528612411540829077943, −7.25520931033063620460080137114, −6.582842633926119601105993552591, −5.85544067003134936804277724884, −4.73591342058923678883410733440, −4.04078646900002887805756561280, −3.08861178364919895225441743799, −2.74873129554942933047016038681, −1.97862620183919674419064693597, −0.741249861055303819468799736992, 0.470937134376607146210182631102, 1.309085367957522753895088331510, 2.11791974894784862600988115479, 2.576870213834228406514078620868, 3.49606444970156756785508016717, 4.999967187578740775883861265099, 5.57460449315734037339368512232, 6.23778667836338460664547419423, 6.72513577362507963403151895067, 7.6147401578635699428603414935, 8.32355373933955523761354717560, 8.80510294726210780325376161855, 9.263098918186075129178651426664, 10.20459995045168126798866338980, 10.57401305473227090500875851378, 11.9368693627597249194605518503, 12.26942469763320323465220632281, 12.793985891848695648431646051014, 13.55482393068981082833180843235, 14.40710148859031022585052492499, 15.06222193677917894920193727055, 15.49274413011468361992237305961, 16.69684556924255298266226269601, 16.92308659192146181681643692322, 17.46894195384229217162511004121

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