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Properties

Label 1-6011-6011.63-r0-0-0
Degree $1$
Conductor $6011$
Sign $0.143 + 0.989i$
Analytic cond. $27.9149$
Root an. cond. $27.9149$
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  + (0.551 − 0.834i)2-s + (0.771 − 0.636i)3-s + (−0.392 − 0.919i)4-s + (0.328 − 0.944i)5-s + (−0.105 − 0.994i)6-s + (−0.999 + 0.0303i)7-s + (−0.983 − 0.179i)8-s + (0.189 − 0.981i)9-s + (−0.606 − 0.794i)10-s + (−0.471 − 0.881i)11-s + (−0.888 − 0.459i)12-s + (0.820 − 0.571i)13-s + (−0.525 + 0.850i)14-s + (−0.347 − 0.937i)15-s + (−0.692 + 0.721i)16-s + (0.212 − 0.977i)17-s + ⋯
L(s)  = 1  + (0.551 − 0.834i)2-s + (0.771 − 0.636i)3-s + (−0.392 − 0.919i)4-s + (0.328 − 0.944i)5-s + (−0.105 − 0.994i)6-s + (−0.999 + 0.0303i)7-s + (−0.983 − 0.179i)8-s + (0.189 − 0.981i)9-s + (−0.606 − 0.794i)10-s + (−0.471 − 0.881i)11-s + (−0.888 − 0.459i)12-s + (0.820 − 0.571i)13-s + (−0.525 + 0.850i)14-s + (−0.347 − 0.937i)15-s + (−0.692 + 0.721i)16-s + (0.212 − 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6011\)
Sign: $0.143 + 0.989i$
Analytic conductor: \(27.9149\)
Root analytic conductor: \(27.9149\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6011} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6011,\ (0:\ ),\ 0.143 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.892287700 - 1.638020196i\)
\(L(\frac12)\) \(\approx\) \(-1.892287700 - 1.638020196i\)
\(L(1)\) \(\approx\) \(0.5112238345 - 1.453502372i\)
\(L(1)\) \(\approx\) \(0.5112238345 - 1.453502372i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6011 \( 1 \)
good2 \( 1 + (0.551 - 0.834i)T \)
3 \( 1 + (0.771 - 0.636i)T \)
5 \( 1 + (0.328 - 0.944i)T \)
7 \( 1 + (-0.999 + 0.0303i)T \)
11 \( 1 + (-0.471 - 0.881i)T \)
13 \( 1 + (0.820 - 0.571i)T \)
17 \( 1 + (0.212 - 0.977i)T \)
19 \( 1 + (-0.496 - 0.867i)T \)
23 \( 1 + (-0.647 - 0.762i)T \)
29 \( 1 + (0.768 + 0.639i)T \)
31 \( 1 + (0.981 + 0.189i)T \)
37 \( 1 + (-0.730 + 0.682i)T \)
41 \( 1 + (-0.998 + 0.0553i)T \)
43 \( 1 + (0.285 - 0.958i)T \)
47 \( 1 + (0.819 - 0.573i)T \)
53 \( 1 + (-0.684 - 0.728i)T \)
59 \( 1 + (0.990 - 0.137i)T \)
61 \( 1 + (-0.132 - 0.991i)T \)
67 \( 1 + (0.0590 + 0.998i)T \)
71 \( 1 + (0.490 - 0.871i)T \)
73 \( 1 + (-0.141 + 0.989i)T \)
79 \( 1 + (0.627 + 0.778i)T \)
83 \( 1 + (0.997 - 0.0668i)T \)
89 \( 1 + (0.991 + 0.133i)T \)
97 \( 1 + (0.994 + 0.106i)T \)
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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

−18.1237708675765091432470300985, −17.409400745605585653010848175651, −16.75623083891880768633120275843, −15.85577174615270414029679430538, −15.66249826884648806615500856217, −14.95422061447862777319474814956, −14.40268606847598153602405036171, −13.67139235991180231712592645886, −13.37696008240985790322183680943, −12.515470987427848047569740266819, −11.81542004755722700183336060825, −10.632612753005101685788682994293, −10.2026377329808712238707059467, −9.5452066665091150900130751066, −8.87435604007991770496949257756, −7.961974958212074401325663860542, −7.58380817565736292018876793318, −6.56329209400081898087046368801, −6.21269879362275791277359225627, −5.45509509072438266196889992913, −4.35770108458171804948097356017, −3.8634223157673319685790585028, −3.29645052374123629773913899970, −2.52424832177972812658508731494, −1.80147470378045695858649773902, 0.54762392244585530278297372684, 0.82621394984725231545815202817, 1.97277950369652132135443465043, 2.69313815749203241923532921643, 3.24626780695107070754744435176, 3.91090593403010096928501515542, 4.90748065429668815247852172133, 5.54782078549261475513910719991, 6.3890531510971488665442509445, 6.799924301592699879638225698467, 8.20661741339074298899775542833, 8.60373838146945462344689933729, 9.14286157193729381397001119555, 9.98571883884658507499617391748, 10.43235988417281529994090555994, 11.48153527724821857863005154646, 12.24965751513458964414597537666, 12.62742054433225758423037068095, 13.38645456806461032984742316818, 13.654176092756971189381235838641, 14.09604790226165219404726546344, 15.29912714094399841478627178518, 15.75428594107075866455144850472, 16.33886530222503445005906475481, 17.44531908112605563925717131822

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