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Properties

Label 1-6017-6017.784-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.599 + 0.800i$
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  + (0.399 + 0.916i)2-s + (−0.809 + 0.587i)3-s + (−0.681 + 0.732i)4-s + (0.958 + 0.285i)5-s + (−0.861 − 0.506i)6-s + (0.215 + 0.976i)7-s + (−0.943 − 0.331i)8-s + (0.309 − 0.951i)9-s + (0.120 + 0.992i)10-s + (0.120 − 0.992i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (−0.943 + 0.331i)15-s + (−0.0724 − 0.997i)16-s + (−0.168 + 0.985i)17-s + (0.995 − 0.0965i)18-s + ⋯
L(s)  = 1  + (0.399 + 0.916i)2-s + (−0.809 + 0.587i)3-s + (−0.681 + 0.732i)4-s + (0.958 + 0.285i)5-s + (−0.861 − 0.506i)6-s + (0.215 + 0.976i)7-s + (−0.943 − 0.331i)8-s + (0.309 − 0.951i)9-s + (0.120 + 0.992i)10-s + (0.120 − 0.992i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (−0.943 + 0.331i)15-s + (−0.0724 − 0.997i)16-s + (−0.168 + 0.985i)17-s + (0.995 − 0.0965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.419681326 + 0.7100859993i\)
\(L(\frac12)\) \(\approx\) \(1.419681326 + 0.7100859993i\)
\(L(1)\) \(\approx\) \(0.8628285401 + 0.6818175826i\)
\(L(1)\) \(\approx\) \(0.8628285401 + 0.6818175826i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.399 + 0.916i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.958 + 0.285i)T \)
7 \( 1 + (0.215 + 0.976i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.168 + 0.985i)T \)
19 \( 1 + (0.644 - 0.764i)T \)
23 \( 1 + (-0.354 - 0.935i)T \)
29 \( 1 + (0.836 - 0.548i)T \)
31 \( 1 + (-0.527 + 0.849i)T \)
37 \( 1 + (0.836 - 0.548i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (-0.262 - 0.964i)T \)
53 \( 1 + (-0.607 - 0.794i)T \)
59 \( 1 + (0.644 + 0.764i)T \)
61 \( 1 + (-0.861 - 0.506i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (-0.527 - 0.849i)T \)
73 \( 1 + (0.0241 - 0.999i)T \)
79 \( 1 + (-0.906 - 0.421i)T \)
83 \( 1 + (0.958 + 0.285i)T \)
89 \( 1 + (-0.748 - 0.663i)T \)
97 \( 1 + (0.715 - 0.698i)T \)
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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

−17.64965744341335305417477648658, −17.22046678835180798922504662108, −16.397700661064152902068876656604, −15.918942067926959329637241072986, −14.476285436108123238458694714749, −14.07799677254577046793979848220, −13.51963411426240285954724600342, −13.11560698653860860466298173352, −12.31736992647754939229968011218, −11.5739967216256279412051077795, −11.234171673135354846647369430126, −10.40913133148336110212203013665, −9.79986044999836150024885839005, −9.325715603680274670613359442173, −8.273739157381034521185309129716, −7.408843631206443854042284398942, −6.62010547716928571765666885864, −5.99122815873028111233122246249, −5.29915582210338477996985871496, −4.66395479533391548178015597578, −4.04509635563127509034319783134, −2.976359648787793660654875967606, −2.10211495647235090989372656040, −1.29902695285217528218003022636, −1.05260522204414275796354929090, 0.42817644456649805591676228757, 1.736916836922794156851383301625, 2.828099967151483682474003988591, 3.41433358946158785831895232162, 4.472333792220339405687822270002, 5.13515052811258091317421207903, 5.5820282742302647646672194362, 6.28416456588075892245781574477, 6.575381262893088687919389205906, 7.65679947892547399434980612658, 8.60382492299828062815493866019, 8.97975008358444566288750385594, 9.86942249519060162572446547465, 10.41272918499808449414914172614, 11.204892552752502606695812145904, 12.05463839331037138892035904523, 12.6200738744898467950744451457, 13.214470520529003390669208701824, 14.00413465752281953488153760845, 14.85431369986972928556089738431, 15.128842411781051682678146431305, 15.82062384162029896476775710554, 16.43817103336459743337318868211, 17.14894198345264951176493983817, 17.78048197653893857089075860710

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