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Properties

Label 1-6017-6017.1949-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.559 - 0.828i$
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  + (−0.877 + 0.478i)2-s + (−0.309 + 0.951i)3-s + (0.541 − 0.840i)4-s + (0.136 − 0.990i)5-s + (−0.184 − 0.982i)6-s + (−0.231 − 0.972i)7-s + (−0.0724 + 0.997i)8-s + (−0.809 − 0.587i)9-s + (0.354 + 0.935i)10-s + (0.632 + 0.774i)12-s + (0.104 − 0.994i)13-s + (0.669 + 0.743i)14-s + (0.899 + 0.435i)15-s + (−0.414 − 0.910i)16-s + (−0.293 + 0.955i)17-s + (0.991 + 0.128i)18-s + ⋯
L(s)  = 1  + (−0.877 + 0.478i)2-s + (−0.309 + 0.951i)3-s + (0.541 − 0.840i)4-s + (0.136 − 0.990i)5-s + (−0.184 − 0.982i)6-s + (−0.231 − 0.972i)7-s + (−0.0724 + 0.997i)8-s + (−0.809 − 0.587i)9-s + (0.354 + 0.935i)10-s + (0.632 + 0.774i)12-s + (0.104 − 0.994i)13-s + (0.669 + 0.743i)14-s + (0.899 + 0.435i)15-s + (−0.414 − 0.910i)16-s + (−0.293 + 0.955i)17-s + (0.991 + 0.128i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.559 - 0.828i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.559 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8429846844 - 0.4479023210i\)
\(L(\frac12)\) \(\approx\) \(0.8429846844 - 0.4479023210i\)
\(L(1)\) \(\approx\) \(0.6702554353 + 0.01200701808i\)
\(L(1)\) \(\approx\) \(0.6702554353 + 0.01200701808i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.877 + 0.478i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.136 - 0.990i)T \)
7 \( 1 + (-0.231 - 0.972i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (-0.293 + 0.955i)T \)
19 \( 1 + (0.993 + 0.112i)T \)
23 \( 1 + (0.845 - 0.534i)T \)
29 \( 1 + (-0.715 - 0.698i)T \)
31 \( 1 + (-0.215 - 0.976i)T \)
37 \( 1 + (-0.247 + 0.968i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.987 - 0.160i)T \)
47 \( 1 + (0.937 + 0.347i)T \)
53 \( 1 + (0.339 - 0.940i)T \)
59 \( 1 + (0.993 - 0.112i)T \)
61 \( 1 + (0.184 + 0.982i)T \)
67 \( 1 + (0.987 - 0.160i)T \)
71 \( 1 + (0.953 - 0.301i)T \)
73 \( 1 + (0.471 - 0.881i)T \)
79 \( 1 + (0.836 - 0.548i)T \)
83 \( 1 + (-0.789 - 0.613i)T \)
89 \( 1 + (-0.568 + 0.822i)T \)
97 \( 1 + (0.513 + 0.857i)T \)
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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

−18.15286666506396023111062584936, −17.38814818944214201552906676246, −16.74745593424581857307765276712, −15.87293987359839850728371153063, −15.49289837728248277185744642350, −14.29450421707595338372231898113, −13.92253258524411168303029194899, −12.990355791548216450579986531947, −12.388696825881759737772965739410, −11.63747275969955312216528745553, −11.34267429223417817248284666572, −10.73600664189794862969641393185, −9.72046509233961965223285489728, −9.13794564531752313143387160165, −8.636507809349796264234938630706, −7.583019499213717458852948371175, −7.000458438269295788294377381122, −6.79167372946091851628428680193, −5.76470108915815904825197452114, −5.14611940755728006986346216653, −3.68015685401850551709973382058, −3.01702366325184070823197141589, −2.339350415126425128755800442376, −1.838374779663461773527245263446, −0.84913861540667563982106146059, 0.50745035922637376010060027260, 0.95400490685117821782579524279, 2.09479058333408672175155946446, 3.241700859088146007703309256843, 4.01282854178743997442145298723, 4.7993676730522408301986553369, 5.47442342589709588273454234873, 6.0004216562240373581564456422, 6.84462569345880406934625688379, 7.77382161483925217274804900856, 8.29512134781093373106518368055, 9.00525743409346254378937101773, 9.71032062974685151739722113941, 10.10744957320227596532577754498, 10.79065508880793964737074935789, 11.36562172763947839936845537796, 12.213664368987951806364660872279, 13.12755488698474715753478499005, 13.69813989022539004128853799428, 14.65493356819993022786301729294, 15.26518466956646924616444230422, 15.77171144874142807265502189828, 16.50826801428093932237599386553, 16.92757257251925647129368634445, 17.33465584393837067256176037115

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