Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.00267 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.00267 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (350, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.00267 - 0.999i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7571146857 - 0.7591461693i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7571146857 - 0.7591461693i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7328067199 - 0.2281520291i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7328067199 - 0.2281520291i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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