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Properties

Label 1-6017-6017.4809-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.470 + 0.882i$
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  + (0.361 − 0.932i)2-s + (0.691 − 0.722i)3-s + (−0.739 − 0.673i)4-s + (−0.147 + 0.989i)5-s + (−0.424 − 0.905i)6-s + (−0.276 − 0.961i)7-s + (−0.894 + 0.446i)8-s + (−0.0448 − 0.998i)9-s + (0.868 + 0.495i)10-s + (−0.997 + 0.0689i)12-s + (0.393 − 0.919i)13-s + (−0.995 − 0.0896i)14-s + (0.612 + 0.790i)15-s + (0.0930 + 0.995i)16-s + (0.897 − 0.440i)17-s + (−0.947 − 0.318i)18-s + ⋯
L(s)  = 1  + (0.361 − 0.932i)2-s + (0.691 − 0.722i)3-s + (−0.739 − 0.673i)4-s + (−0.147 + 0.989i)5-s + (−0.424 − 0.905i)6-s + (−0.276 − 0.961i)7-s + (−0.894 + 0.446i)8-s + (−0.0448 − 0.998i)9-s + (0.868 + 0.495i)10-s + (−0.997 + 0.0689i)12-s + (0.393 − 0.919i)13-s + (−0.995 − 0.0896i)14-s + (0.612 + 0.790i)15-s + (0.0930 + 0.995i)16-s + (0.897 − 0.440i)17-s + (−0.947 − 0.318i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.470 + 0.882i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (4809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.470 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2151404344 - 0.1290708957i\)
\(L(\frac12)\) \(\approx\) \(-0.2151404344 - 0.1290708957i\)
\(L(1)\) \(\approx\) \(0.7844100939 - 0.7926906233i\)
\(L(1)\) \(\approx\) \(0.7844100939 - 0.7926906233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.361 - 0.932i)T \)
3 \( 1 + (0.691 - 0.722i)T \)
5 \( 1 + (-0.147 + 0.989i)T \)
7 \( 1 + (-0.276 - 0.961i)T \)
13 \( 1 + (0.393 - 0.919i)T \)
17 \( 1 + (0.897 - 0.440i)T \)
19 \( 1 + (-0.436 + 0.899i)T \)
23 \( 1 + (-0.449 - 0.893i)T \)
29 \( 1 + (0.824 + 0.565i)T \)
31 \( 1 + (-0.655 + 0.755i)T \)
37 \( 1 + (-0.734 + 0.678i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.154 + 0.987i)T \)
47 \( 1 + (-0.527 + 0.849i)T \)
53 \( 1 + (-0.960 - 0.279i)T \)
59 \( 1 + (-0.779 + 0.626i)T \)
61 \( 1 + (-0.977 + 0.212i)T \)
67 \( 1 + (0.997 - 0.0689i)T \)
71 \( 1 + (-0.590 + 0.806i)T \)
73 \( 1 + (0.0310 - 0.999i)T \)
79 \( 1 + (0.705 - 0.708i)T \)
83 \( 1 + (-0.681 - 0.732i)T \)
89 \( 1 + (-0.915 + 0.402i)T \)
97 \( 1 + (-0.315 - 0.948i)T \)
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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

−18.17444107934697175254426851003, −17.25051348741638547042685402558, −16.71827015642159268460398254723, −16.161591088635998227008549752420, −15.49202150577734020773293978455, −15.31467396337325908707097908279, −14.368371680459356045415925372840, −13.79854263911060420990065607268, −13.18010540586394481576295574498, −12.50354579295226403444366077568, −11.87510033707949066315777834839, −11.093625830691885607837130109219, −9.78230315591083684915803315783, −9.4314888029945145265226765576, −8.84665216458265606730716065730, −8.27148723828165208877884330965, −7.77863001429961162364228360215, −6.75919525600104885667403459998, −5.89312303485900755308012693111, −5.33893668448245397155117245140, −4.68813495427929934927281092612, −3.97354223201796460933482885163, −3.44177399980589017280102644715, −2.45781754044563327675329492407, −1.56672825540673038268180296788, 0.04889209080055815714253260056, 1.14163746946837820566132199531, 1.73554473515313608376744328622, 2.9101496595079521918018229587, 3.12033851614141465463647409661, 3.76906202060134961179519968897, 4.57419563597941751827835456046, 5.74107007560457653011205894125, 6.36430841889541800679700547074, 7.037587121731051984569093600506, 7.90128692841973624614141761977, 8.322281943205201088552787029674, 9.36716803158813563262960332972, 10.120040414892755976367654929755, 10.510796938142103293361379798603, 11.12166913587761968288957689389, 12.193219231871352378309267116852, 12.45660962062433127680737183644, 13.238094397597378597370680148380, 13.999763769609144589035879329126, 14.25985054527162182641680512360, 14.80100647749613469358740151906, 15.65053598514472171821142250926, 16.510829190901731402160075409902, 17.699827582289844982457288571116

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