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Properties

Label 1-6017-6017.138-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.847 + 0.531i$
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  + (0.161 + 0.986i)2-s + (−0.134 + 0.990i)3-s + (−0.947 + 0.318i)4-s + (−0.696 − 0.718i)5-s + (−0.999 + 0.0276i)6-s + (−0.367 − 0.930i)7-s + (−0.467 − 0.883i)8-s + (−0.963 − 0.266i)9-s + (0.596 − 0.802i)10-s + (−0.188 − 0.982i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.804 − 0.593i)15-s + (0.796 − 0.604i)16-s + (0.949 + 0.312i)17-s + (0.106 − 0.994i)18-s + ⋯
L(s)  = 1  + (0.161 + 0.986i)2-s + (−0.134 + 0.990i)3-s + (−0.947 + 0.318i)4-s + (−0.696 − 0.718i)5-s + (−0.999 + 0.0276i)6-s + (−0.367 − 0.930i)7-s + (−0.467 − 0.883i)8-s + (−0.963 − 0.266i)9-s + (0.596 − 0.802i)10-s + (−0.188 − 0.982i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.804 − 0.593i)15-s + (0.796 − 0.604i)16-s + (0.949 + 0.312i)17-s + (0.106 − 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.847 + 0.531i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.847 + 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6337133945 + 0.1822882689i\)
\(L(\frac12)\) \(\approx\) \(0.6337133945 + 0.1822882689i\)
\(L(1)\) \(\approx\) \(0.5910668910 + 0.3713084511i\)
\(L(1)\) \(\approx\) \(0.5910668910 + 0.3713084511i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.161 + 0.986i)T \)
3 \( 1 + (-0.134 + 0.990i)T \)
5 \( 1 + (-0.696 - 0.718i)T \)
7 \( 1 + (-0.367 - 0.930i)T \)
13 \( 1 + (-0.753 - 0.657i)T \)
17 \( 1 + (0.949 + 0.312i)T \)
19 \( 1 + (-0.0655 + 0.997i)T \)
23 \( 1 + (0.994 - 0.103i)T \)
29 \( 1 + (-0.762 + 0.647i)T \)
31 \( 1 + (-0.705 - 0.708i)T \)
37 \( 1 + (-0.461 - 0.887i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.999 - 0.0345i)T \)
47 \( 1 + (-0.906 + 0.421i)T \)
53 \( 1 + (-0.999 + 0.00690i)T \)
59 \( 1 + (-0.958 + 0.285i)T \)
61 \( 1 + (-0.195 + 0.980i)T \)
67 \( 1 + (0.188 + 0.982i)T \)
71 \( 1 + (0.847 + 0.530i)T \)
73 \( 1 + (0.302 + 0.953i)T \)
79 \( 1 + (0.828 - 0.559i)T \)
83 \( 1 + (0.995 - 0.0965i)T \)
89 \( 1 + (0.418 - 0.908i)T \)
97 \( 1 + (-0.882 + 0.470i)T \)
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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

−18.01741632562604629945225806287, −17.17954604620730086370733834316, −16.54966398372583394933702512806, −15.36316783613750370305926100900, −14.94911999274065473471681840208, −14.19481747508889221141030793280, −13.65407324648152549459750778595, −12.77780967890065031170266966215, −12.30204984525545619577658177411, −11.79515152745963088452484962760, −11.25562163514361535752932300513, −10.675011743400350112627054394273, −9.581591364687730387409753931788, −9.1655127946621717623892604344, −8.285854491947019080970083438146, −7.630243263128900933156188083666, −6.81099962049861269405938466549, −6.27155124131150562904769779840, −5.17024281723474478331613213873, −4.87509470626491379358569868917, −3.3730492634628232604312160463, −3.21812383768609875137758174020, −2.28830982082984134038707084978, −1.78954551645233670099357401638, −0.55432079941445424925430290233, 0.29992490755458074689998891661, 1.2856340182528214011736553791, 3.11633034597398217766905532471, 3.56691660736718999285427139339, 4.16885250602647317353903250140, 4.91978850538651607704487695578, 5.3957219377872906795097203008, 6.13002269198777220564962842677, 7.14020114236011510022218353327, 7.76457191137325176922035560887, 8.2211268739971564618959819654, 9.16159634613271767909975943308, 9.64317118241532639754375938173, 10.31104822754838841917762358331, 11.045384477519864655561948150882, 11.97837060122822989333788317677, 12.7381702675617827096147117019, 13.06497694169974615167966984089, 14.1670162114685157905876106037, 14.76023501608507102479630039141, 15.06474544728408205497961106275, 16.07419999211145692109571464639, 16.34262380044297492748546401031, 17.04796568486070584813873720284, 17.13557518448031618671033402536

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