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Properties

Label 1-6027-6027.872-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.653 - 0.756i$
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  + (−0.999 + 0.0149i)2-s + (0.999 − 0.0299i)4-s + (−0.701 − 0.712i)5-s + (−0.998 + 0.0448i)8-s + (0.712 + 0.701i)10-s + (0.786 + 0.617i)11-s + (0.969 + 0.244i)13-s + (0.998 − 0.0598i)16-s + (0.229 + 0.973i)17-s + (0.998 + 0.0523i)19-s + (−0.722 − 0.691i)20-s + (−0.795 − 0.605i)22-s + (0.971 − 0.237i)23-s + (−0.0149 + 0.999i)25-s + (−0.973 − 0.229i)26-s + ⋯
L(s)  = 1  + (−0.999 + 0.0149i)2-s + (0.999 − 0.0299i)4-s + (−0.701 − 0.712i)5-s + (−0.998 + 0.0448i)8-s + (0.712 + 0.701i)10-s + (0.786 + 0.617i)11-s + (0.969 + 0.244i)13-s + (0.998 − 0.0598i)16-s + (0.229 + 0.973i)17-s + (0.998 + 0.0523i)19-s + (−0.722 − 0.691i)20-s + (−0.795 − 0.605i)22-s + (0.971 − 0.237i)23-s + (−0.0149 + 0.999i)25-s + (−0.973 − 0.229i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.653 - 0.756i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (872, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.653 - 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9958640545 - 0.4556901267i\)
\(L(\frac12)\) \(\approx\) \(0.9958640545 - 0.4556901267i\)
\(L(1)\) \(\approx\) \(0.7233587239 - 0.08283037138i\)
\(L(1)\) \(\approx\) \(0.7233587239 - 0.08283037138i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.999 + 0.0149i)T \)
5 \( 1 + (-0.701 - 0.712i)T \)
11 \( 1 + (0.786 + 0.617i)T \)
13 \( 1 + (0.969 + 0.244i)T \)
17 \( 1 + (0.229 + 0.973i)T \)
19 \( 1 + (0.998 + 0.0523i)T \)
23 \( 1 + (0.971 - 0.237i)T \)
29 \( 1 + (-0.822 - 0.569i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.337 + 0.941i)T \)
43 \( 1 + (0.351 - 0.936i)T \)
47 \( 1 + (-0.717 - 0.696i)T \)
53 \( 1 + (-0.727 - 0.685i)T \)
59 \( 1 + (0.163 + 0.986i)T \)
61 \( 1 + (-0.959 + 0.280i)T \)
67 \( 1 + (0.933 - 0.358i)T \)
71 \( 1 + (-0.569 - 0.822i)T \)
73 \( 1 + (-0.294 - 0.955i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.961 + 0.273i)T \)
97 \( 1 + (0.156 - 0.987i)T \)
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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

−17.94021874312005707462190464194, −17.25522160259920795697379939509, −16.337896967109883475859772756784, −15.9770170060650733186133450309, −15.49484758511414287169546465517, −14.44228303165304990597561321293, −14.23654935223535144257764442652, −13.13392654344019856446292067459, −12.242014538249757591426075888441, −11.60269775020998973316830394440, −11.00227483992089864700055564919, −10.79991877405693026639051650231, −9.62671493213744350243918791417, −9.206081358679668410651546562714, −8.45313924432374977836626430715, −7.75198858054483267397531332614, −7.14664740586630722459362607190, −6.58390116341610541778584509262, −5.837835194583934978659578549911, −4.96751794363238872981216039895, −3.63565215511684692481394928447, −3.32523945409899199167373448722, −2.61981833668512006470366478112, −1.3619930377915407127592662825, −0.842751130079362109763648800207, 0.53271490880149673880020925506, 1.41142372170903377382362002818, 1.87444591507826678000849768900, 3.21819000050156749896741386880, 3.739637779820298272827174729987, 4.58989003454628448749770884260, 5.55278569012865949595767649771, 6.28507401363855735594902638822, 7.01834684980663846042272378385, 7.7347273827863888265678056699, 8.26791316432348506412970270448, 9.05081856592326548263159259589, 9.37471394577398286503737993997, 10.27033857166592588311311924516, 11.00973127698217292906241431594, 11.71869928506439831163993268340, 12.024033959977128876948354694787, 12.910734079466072074054471623251, 13.55303884406288026328311445175, 14.71185205887541540445314034822, 15.18405864798965187636389876775, 15.71943621096373738052920465590, 16.63179527344017864952284793394, 16.810971420609201818452868962864, 17.536314174215072245481261056042

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