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Properties

Label 1-6017-6017.1934-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.332 + 0.943i$
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  + (0.836 − 0.548i)2-s + (0.309 − 0.951i)3-s + (0.399 − 0.916i)4-s + (−0.989 + 0.144i)5-s + (−0.262 − 0.964i)6-s + (0.779 − 0.626i)7-s + (−0.168 − 0.985i)8-s + (−0.809 − 0.587i)9-s + (−0.748 + 0.663i)10-s + (−0.748 − 0.663i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.168 + 0.985i)15-s + (−0.681 − 0.732i)16-s + (0.644 − 0.764i)17-s + (−0.998 − 0.0483i)18-s + ⋯
L(s)  = 1  + (0.836 − 0.548i)2-s + (0.309 − 0.951i)3-s + (0.399 − 0.916i)4-s + (−0.989 + 0.144i)5-s + (−0.262 − 0.964i)6-s + (0.779 − 0.626i)7-s + (−0.168 − 0.985i)8-s + (−0.809 − 0.587i)9-s + (−0.748 + 0.663i)10-s + (−0.748 − 0.663i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.168 + 0.985i)15-s + (−0.681 − 0.732i)16-s + (0.644 − 0.764i)17-s + (−0.998 − 0.0483i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.332 + 0.943i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1934, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.332 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.254506673 - 1.773002732i\)
\(L(\frac12)\) \(\approx\) \(-1.254506673 - 1.773002732i\)
\(L(1)\) \(\approx\) \(0.8185424158 - 1.271685446i\)
\(L(1)\) \(\approx\) \(0.8185424158 - 1.271685446i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.836 - 0.548i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.989 + 0.144i)T \)
7 \( 1 + (0.779 - 0.626i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.644 - 0.764i)T \)
19 \( 1 + (-0.906 - 0.421i)T \)
23 \( 1 + (0.568 + 0.822i)T \)
29 \( 1 + (0.958 + 0.285i)T \)
31 \( 1 + (0.485 - 0.873i)T \)
37 \( 1 + (0.958 + 0.285i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.748 - 0.663i)T \)
47 \( 1 + (-0.607 - 0.794i)T \)
53 \( 1 + (-0.443 - 0.896i)T \)
59 \( 1 + (-0.906 + 0.421i)T \)
61 \( 1 + (-0.262 - 0.964i)T \)
67 \( 1 + (-0.748 - 0.663i)T \)
71 \( 1 + (0.485 + 0.873i)T \)
73 \( 1 + (0.715 + 0.698i)T \)
79 \( 1 + (0.215 + 0.976i)T \)
83 \( 1 + (-0.989 + 0.144i)T \)
89 \( 1 + (-0.354 - 0.935i)T \)
97 \( 1 + (0.926 + 0.377i)T \)
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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

−17.96745310157651460263923516328, −17.16860293266166857788962854877, −16.52573991199600084013024174232, −16.2331871039615336094885845154, −15.203793493120411799002915220272, −15.009995712614325047360319422318, −14.51036016379926236849439854260, −13.93943117231803803489861149169, −12.75900981900286409151005249025, −12.30054465776693277262666635815, −11.67963855449960566981437426305, −11.03245694158554755725516562507, −10.41044845222382016438438765672, −9.31102730209519160875232613750, −8.5909173894930665179527778436, −8.09441767305982092266776164081, −7.66594703663479372940436809199, −6.540093312469541833056922962099, −5.91928885903345251493029102114, −4.80878730688476240693247934814, −4.670289373267388446871471009371, −4.09151391773472655289783950605, −3.06461400708845978975135235409, −2.66411090018804763128151461972, −1.55658639477282147888613452059, 0.40257479478060314504918649156, 1.020225973853745378171471204626, 1.98825191356145195406454181438, 2.7780723321492944159592682098, 3.3364982829655455601232950862, 4.180521632218896901747804825976, 4.897564715394978924325542607048, 5.50689351571450252101532314930, 6.66365757787086498704259761513, 7.05641385124430737205660353169, 7.786596956592254066063059416775, 8.266671108331668523991977883564, 9.31407716126478017476667090789, 10.1723920533373958016039162757, 10.93771966402602377154847371519, 11.554416610566094612393895203766, 11.94330699530709576904931041455, 12.68774649112654919214270150835, 13.21734184442162943874409503795, 14.028957392227564841499164751503, 14.427906497789105115669708654208, 15.17838854051286280731815841882, 15.4944202478437154849565703357, 16.71432597576252747176609143864, 17.2564767010999367868890348738

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