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Properties

Label 1-6027-6027.3440-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.715 + 0.698i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.575 + 0.817i)2-s + (−0.337 − 0.941i)4-s + (0.887 + 0.460i)5-s + (0.963 + 0.266i)8-s + (−0.887 + 0.460i)10-s + (−0.193 + 0.981i)11-s + (0.995 + 0.0896i)13-s + (−0.772 + 0.635i)16-s + (−0.646 − 0.762i)17-s + (0.978 + 0.207i)19-s + (0.134 − 0.990i)20-s + (−0.691 − 0.722i)22-s + (0.925 + 0.379i)23-s + (0.575 + 0.817i)25-s + (−0.646 + 0.762i)26-s + ⋯
L(s)  = 1  + (−0.575 + 0.817i)2-s + (−0.337 − 0.941i)4-s + (0.887 + 0.460i)5-s + (0.963 + 0.266i)8-s + (−0.887 + 0.460i)10-s + (−0.193 + 0.981i)11-s + (0.995 + 0.0896i)13-s + (−0.772 + 0.635i)16-s + (−0.646 − 0.762i)17-s + (0.978 + 0.207i)19-s + (0.134 − 0.990i)20-s + (−0.691 − 0.722i)22-s + (0.925 + 0.379i)23-s + (0.575 + 0.817i)25-s + (−0.646 + 0.762i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.715 + 0.698i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (3440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.715 + 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5247963212 + 1.288887247i\)
\(L(\frac12)\) \(\approx\) \(0.5247963212 + 1.288887247i\)
\(L(1)\) \(\approx\) \(0.8068226667 + 0.4697333426i\)
\(L(1)\) \(\approx\) \(0.8068226667 + 0.4697333426i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.575 + 0.817i)T \)
5 \( 1 + (0.887 + 0.460i)T \)
11 \( 1 + (-0.193 + 0.981i)T \)
13 \( 1 + (0.995 + 0.0896i)T \)
17 \( 1 + (-0.646 - 0.762i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.925 + 0.379i)T \)
29 \( 1 + (-0.473 - 0.880i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.999 - 0.0299i)T \)
43 \( 1 + (-0.550 - 0.834i)T \)
47 \( 1 + (0.575 - 0.817i)T \)
53 \( 1 + (0.337 + 0.941i)T \)
59 \( 1 + (-0.447 + 0.894i)T \)
61 \( 1 + (-0.791 - 0.611i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (-0.473 + 0.880i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.420 - 0.907i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−17.63978536844880267549886480371, −16.8940702615903348454731810923, −16.31864356441906153815241523232, −15.85395886430340137115252006570, −14.70712839075802065131519670722, −13.86628690147797714968959935223, −13.36037954312449189762638814376, −12.91814957672552263714479881854, −12.22340742189679388703600039454, −11.24929072774645452491195461786, −10.837384973940324682346544253, −10.30799598491349023395158065706, −9.337659124212651126714368071338, −8.85949244688353899012920794712, −8.495988098622894227778636697165, −7.56067527931390276579100738688, −6.68263742849618352224173771236, −5.906505046292656899722617695298, −5.1620932456063903652935750449, −4.41036728435099626496502708636, −3.281484010922361061139546926936, −3.09081154340663772987984493278, −1.78743117464085941339640968806, −1.448309082475901042755590488611, −0.4594760228315411778738050075, 1.01714444497859804368541771681, 1.78230891600993928872122600929, 2.48817976388922363419738090787, 3.572477759230841467483062465781, 4.51676587530010814389191451147, 5.40119100122281607846966202294, 5.719004558325877890939457593507, 6.68737618796688141896512716337, 7.16933754055928913824636540785, 7.679087233279237096071564496697, 8.9163474619339363194728393693, 9.09399586139333884139491793454, 9.92874837910173341646766909482, 10.433496920582896421118165754917, 11.16096848263489456783404495184, 11.86978618409305835642755167503, 13.169457647463399297896990328297, 13.40365230381660579593945415543, 14.117694742587085053551219484983, 14.765226774180013499051999727408, 15.550099451202688112611006830746, 15.79154821103850418603709029926, 16.9269806006962213135201429260, 17.18994832979376936710113666340, 18.00964690353028193478025529193

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