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Properties

Label 1-6013-6013.79-r0-0-0
Degree $1$
Conductor $6013$
Sign $-0.435 - 0.900i$
Analytic cond. $27.9242$
Root an. cond. $27.9242$
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  + (−0.643 − 0.765i)2-s + (0.999 + 0.0292i)3-s + (−0.171 + 0.985i)4-s + (−0.967 − 0.253i)5-s + (−0.621 − 0.783i)6-s + (0.864 − 0.503i)8-s + (0.998 + 0.0585i)9-s + (0.428 + 0.903i)10-s + (−0.998 − 0.0512i)11-s + (−0.200 + 0.979i)12-s + (0.415 + 0.909i)13-s + (−0.959 − 0.281i)15-s + (−0.941 − 0.337i)16-s + (0.961 − 0.274i)17-s + (−0.597 − 0.801i)18-s + (−0.786 + 0.618i)19-s + ⋯
L(s)  = 1  + (−0.643 − 0.765i)2-s + (0.999 + 0.0292i)3-s + (−0.171 + 0.985i)4-s + (−0.967 − 0.253i)5-s + (−0.621 − 0.783i)6-s + (0.864 − 0.503i)8-s + (0.998 + 0.0585i)9-s + (0.428 + 0.903i)10-s + (−0.998 − 0.0512i)11-s + (−0.200 + 0.979i)12-s + (0.415 + 0.909i)13-s + (−0.959 − 0.281i)15-s + (−0.941 − 0.337i)16-s + (0.961 − 0.274i)17-s + (−0.597 − 0.801i)18-s + (−0.786 + 0.618i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6013\)    =    \(7 \cdot 859\)
Sign: $-0.435 - 0.900i$
Analytic conductor: \(27.9242\)
Root analytic conductor: \(27.9242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6013} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6013,\ (0:\ ),\ -0.435 - 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4915280740 - 0.7834881744i\)
\(L(\frac12)\) \(\approx\) \(0.4915280740 - 0.7834881744i\)
\(L(1)\) \(\approx\) \(0.7689224990 - 0.2671333083i\)
\(L(1)\) \(\approx\) \(0.7689224990 - 0.2671333083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
859 \( 1 \)
good2 \( 1 + (-0.643 - 0.765i)T \)
3 \( 1 + (0.999 + 0.0292i)T \)
5 \( 1 + (-0.967 - 0.253i)T \)
11 \( 1 + (-0.998 - 0.0512i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.961 - 0.274i)T \)
19 \( 1 + (-0.786 + 0.618i)T \)
23 \( 1 + (-0.996 - 0.0804i)T \)
29 \( 1 + (-0.549 - 0.835i)T \)
31 \( 1 + (-0.977 + 0.210i)T \)
37 \( 1 + (-0.0841 - 0.996i)T \)
41 \( 1 + (0.975 + 0.217i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.977 - 0.210i)T \)
53 \( 1 + (-0.728 - 0.684i)T \)
59 \( 1 + (0.933 - 0.358i)T \)
61 \( 1 + (-0.327 + 0.945i)T \)
67 \( 1 + (0.105 - 0.994i)T \)
71 \( 1 + (-0.586 - 0.810i)T \)
73 \( 1 + (-0.537 + 0.843i)T \)
79 \( 1 + (-0.525 + 0.851i)T \)
83 \( 1 + (0.375 - 0.927i)T \)
89 \( 1 + (0.957 - 0.288i)T \)
97 \( 1 + (0.790 - 0.612i)T \)
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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

−18.02704090488262793911333082213, −17.36675656938552469250800107986, −16.19708214011461502110931511212, −16.02217849530854811217689308357, −15.32403987449663866401510793411, −14.760191628872380651082444319992, −14.392869939638451888905883581060, −13.35065967677501017608756555399, −12.92047691504105514802149265193, −12.07740125714458702222678316790, −10.88529514065996293834667180446, −10.585054018495906125949778949823, −9.87764843573162141716288425411, −9.025320073106241309228194286456, −8.37361686400927560305096807826, −7.85802711203863666450682742401, −7.517840529434709812245935948348, −6.748078156164519996189790952381, −5.82836438468171809838122483499, −5.06477018025204108143591762321, −4.20951788131504057076071448528, −3.46841618796281372007468395626, −2.704739970026670811828985033202, −1.80901361506446831512288036617, −0.79918609319869921569346478831, 0.31864764316672470648875987460, 1.499963025000811038429854109684, 2.10836835501389397042364311820, 2.9480389415317892668627267973, 3.70229827753279095933252901455, 4.10265363522786306412147076886, 4.87018065828760428132183510896, 6.09540135854809569515554534191, 7.257185327847553499049681541940, 7.68706481730155423244807569451, 8.19527297926743006980458929147, 8.77804646201229665154522845885, 9.51198973884147133936405384494, 10.09205947684046431099207320272, 10.87680695754967749094050528847, 11.46942765333912329613385616647, 12.29937196864676166126192263770, 12.78889191053262694372418624945, 13.35467537255297598711025081156, 14.29328522911998474006580757097, 14.746732612586838810392067667948, 15.84908235413963211090251118931, 16.17439968669752147651171726943, 16.6301147257287795465616147496, 17.80149202244143827533741212217

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