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Properties

Label 1-6017-6017.120-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.843 - 0.537i$
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  + (−0.733 − 0.680i)2-s + (0.900 + 0.433i)3-s + (0.0747 + 0.997i)4-s + (−0.365 + 0.930i)5-s + (−0.365 − 0.930i)6-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.900 − 0.433i)10-s + (−0.365 + 0.930i)12-s + (−0.955 − 0.294i)13-s + (−0.733 + 0.680i)14-s + (−0.733 + 0.680i)15-s + (−0.988 + 0.149i)16-s + (−0.988 + 0.149i)17-s + (0.0747 − 0.997i)18-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)2-s + (0.900 + 0.433i)3-s + (0.0747 + 0.997i)4-s + (−0.365 + 0.930i)5-s + (−0.365 − 0.930i)6-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.900 − 0.433i)10-s + (−0.365 + 0.930i)12-s + (−0.955 − 0.294i)13-s + (−0.733 + 0.680i)14-s + (−0.733 + 0.680i)15-s + (−0.988 + 0.149i)16-s + (−0.988 + 0.149i)17-s + (0.0747 − 0.997i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.843 - 0.537i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.843 - 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.320184258 - 0.3848648986i\)
\(L(\frac12)\) \(\approx\) \(1.320184258 - 0.3848648986i\)
\(L(1)\) \(\approx\) \(0.9157486953 - 0.08644162089i\)
\(L(1)\) \(\approx\) \(0.9157486953 - 0.08644162089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.733 - 0.680i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
7 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (-0.955 - 0.294i)T \)
17 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (0.733 - 0.680i)T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + (0.900 + 0.433i)T \)
37 \( 1 + (0.988 - 0.149i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.0747 - 0.997i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.955 + 0.294i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (-0.988 + 0.149i)T \)
71 \( 1 + (0.988 - 0.149i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.900 + 0.433i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (-0.733 + 0.680i)T \)
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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

−17.83275012099368962803612188126, −17.118301442699972108025400642292, −16.467430693724916717447257318509, −15.633566224746277309855872333894, −15.29865538390737188132719826029, −14.74928188147456891235138304337, −13.9215470510224636803013910632, −13.27154153132355327140205835602, −12.60604589859079571553685115423, −11.73163298822630754149098351467, −11.40996233428697409729585753105, −9.941207481789299073702162981482, −9.55848888233056590836212201422, −8.93536435336221182763245731250, −8.5190969633365454922745827157, −7.6278261200156369478035066166, −7.46040414497647615812442799432, −6.361455574355370353928988576684, −5.737194871771362594808719716335, −4.84340693056395064795761235624, −4.31750533631255199028337628577, −3.08671894089235779402943558885, −2.235566687117027279609127958707, −1.65067296496666886989915826889, −0.73272753096375817482945021031, 0.5256725812301266392683567841, 1.64657555376631153589580706299, 2.59661586641007691549647169335, 2.92871572919812395438200725968, 3.734935592400394116193990150117, 4.35999138751057709655122241617, 5.03892310151546137169380990965, 6.833490860585561908291380971997, 6.965510443923473649059588981095, 7.71386621186852040383712860074, 8.328273785207714539665900127429, 9.07118323695255114149943151192, 9.871509742313218830137076895332, 10.23026947973036455429701436950, 10.98155946353351493022744888205, 11.30801806343672476794409303565, 12.34357758317251638187510887642, 13.14369804362592722118442710546, 13.67205470707507654418463231054, 14.360714561046574001911687602336, 15.13845472754804238596280646067, 15.57136263359584870739387165331, 16.532519961163926650353127159082, 16.96036014610571394424849846264, 17.903051863079347726677211840692

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