Properties

Label 1-6017-6017.512-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.457 + 0.889i$
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  + (0.865 − 0.500i)2-s + (−0.134 + 0.990i)3-s + (0.498 − 0.867i)4-s + (−0.949 − 0.312i)5-s + (0.380 + 0.924i)6-s + (0.999 − 0.0138i)7-s + (−0.00345 − 0.999i)8-s + (−0.963 − 0.266i)9-s + (−0.978 + 0.205i)10-s + (0.792 + 0.609i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.436 − 0.899i)15-s + (−0.503 − 0.863i)16-s + (−0.847 − 0.530i)17-s + (−0.967 + 0.252i)18-s + ⋯
L(s)  = 1  + (0.865 − 0.500i)2-s + (−0.134 + 0.990i)3-s + (0.498 − 0.867i)4-s + (−0.949 − 0.312i)5-s + (0.380 + 0.924i)6-s + (0.999 − 0.0138i)7-s + (−0.00345 − 0.999i)8-s + (−0.963 − 0.266i)9-s + (−0.978 + 0.205i)10-s + (0.792 + 0.609i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.436 − 0.899i)15-s + (−0.503 − 0.863i)16-s + (−0.847 − 0.530i)17-s + (−0.967 + 0.252i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.457 + 0.889i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.457 + 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1093862759 + 0.1793096183i\)
\(L(\frac12)\) \(\approx\) \(0.1093862759 + 0.1793096183i\)
\(L(1)\) \(\approx\) \(1.121445506 - 0.2406331864i\)
\(L(1)\) \(\approx\) \(1.121445506 - 0.2406331864i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.865 - 0.500i)T \)
3 \( 1 + (-0.134 + 0.990i)T \)
5 \( 1 + (-0.949 - 0.312i)T \)
7 \( 1 + (0.999 - 0.0138i)T \)
13 \( 1 + (-0.753 - 0.657i)T \)
17 \( 1 + (-0.847 - 0.530i)T \)
19 \( 1 + (-0.998 + 0.0552i)T \)
23 \( 1 + (0.479 - 0.877i)T \)
29 \( 1 + (-0.965 - 0.259i)T \)
31 \( 1 + (-0.788 + 0.615i)T \)
37 \( 1 + (0.467 - 0.883i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.770 - 0.636i)T \)
47 \( 1 + (-0.861 - 0.506i)T \)
53 \( 1 + (-0.882 + 0.470i)T \)
59 \( 1 + (0.168 + 0.985i)T \)
61 \( 1 + (0.986 - 0.164i)T \)
67 \( 1 + (-0.792 - 0.609i)T \)
71 \( 1 + (-0.424 + 0.905i)T \)
73 \( 1 + (-0.175 + 0.984i)T \)
79 \( 1 + (-0.249 + 0.968i)T \)
83 \( 1 + (0.836 - 0.548i)T \)
89 \( 1 + (0.700 + 0.713i)T \)
97 \( 1 + (0.969 - 0.246i)T \)
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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

−17.52919772152004759184498589885, −16.8336081895832129768394493678, −16.2100050711048674467327589421, −15.19335550200396744301630298655, −14.79758133154910968797301799770, −14.41336344811591116387103174781, −13.52319789515892622102242640545, −12.88641462008625464688218184958, −12.38013451881442744247133753636, −11.480003804071092056130123559038, −11.34744099966334017235145996865, −10.715927475080945209951026851797, −9.12463275251308280767036104753, −8.51445244643812207610075391262, −7.690560458065527718687364400593, −7.51100383447052445914626816843, −6.7110350322905840184037335271, −6.13633480964097762723544063266, −5.20320823727951546895951897292, −4.56749682400953162395251218091, −3.9313525632791182975792315800, −3.014582567825526044710090278420, −2.134921645178015048226212064560, −1.65568827672159803594866022069, −0.04057908698462911344709910891, 0.94773087065985805783512693951, 2.222156223521370445728682354993, 2.768292364334633742363933693471, 3.81963207201993221573089854096, 4.20707483243086122978279360719, 4.95218384207230743919751718385, 5.20730514731335925666230104069, 6.203678054637539804562737977283, 7.15317077666261226605011128072, 7.86720919714054250720755636987, 8.77010062817456909096856003636, 9.27025280904879553048267835624, 10.32518154293006471099785996451, 10.84368296164066943073940467374, 11.28852539762083205826061371104, 11.866380224657407359862465852347, 12.63513236512365126384648268034, 13.14791701804687083591016412807, 14.32737084936738622320727943028, 14.7249179912050705257263564144, 15.09842505017861331701747237100, 15.79568307783595864094784064999, 16.40818010721024056959971079877, 17.10407793092492847875468213758, 17.885294296984842433191611040019

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