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Properties

Label 1-6027-6027.53-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.985 - 0.170i$
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  + (0.0598 + 0.998i)2-s + (−0.992 + 0.119i)4-s + (−0.0299 + 0.999i)5-s + (−0.178 − 0.983i)8-s + (−0.999 + 0.0299i)10-s + (0.953 + 0.301i)11-s + (0.200 − 0.979i)13-s + (0.971 − 0.237i)16-s + (−0.141 − 0.989i)17-s + (−0.838 + 0.544i)19-s + (−0.0896 − 0.995i)20-s + (−0.244 + 0.969i)22-s + (0.575 − 0.817i)23-s + (−0.998 − 0.0598i)25-s + (0.989 + 0.141i)26-s + ⋯
L(s)  = 1  + (0.0598 + 0.998i)2-s + (−0.992 + 0.119i)4-s + (−0.0299 + 0.999i)5-s + (−0.178 − 0.983i)8-s + (−0.999 + 0.0299i)10-s + (0.953 + 0.301i)11-s + (0.200 − 0.979i)13-s + (0.971 − 0.237i)16-s + (−0.141 − 0.989i)17-s + (−0.838 + 0.544i)19-s + (−0.0896 − 0.995i)20-s + (−0.244 + 0.969i)22-s + (0.575 − 0.817i)23-s + (−0.998 − 0.0598i)25-s + (0.989 + 0.141i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.985 - 0.170i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.985 - 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9014899526 - 0.07737508805i\)
\(L(\frac12)\) \(\approx\) \(0.9014899526 - 0.07737508805i\)
\(L(1)\) \(\approx\) \(0.7823864976 + 0.4471144624i\)
\(L(1)\) \(\approx\) \(0.7823864976 + 0.4471144624i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.0598 + 0.998i)T \)
5 \( 1 + (-0.0299 + 0.999i)T \)
11 \( 1 + (0.953 + 0.301i)T \)
13 \( 1 + (0.200 - 0.979i)T \)
17 \( 1 + (-0.141 - 0.989i)T \)
19 \( 1 + (-0.838 + 0.544i)T \)
23 \( 1 + (0.575 - 0.817i)T \)
29 \( 1 + (-0.997 - 0.0672i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.193 + 0.981i)T \)
43 \( 1 + (0.990 - 0.134i)T \)
47 \( 1 + (-0.663 + 0.748i)T \)
53 \( 1 + (0.786 + 0.617i)T \)
59 \( 1 + (-0.791 + 0.611i)T \)
61 \( 1 + (-0.907 - 0.420i)T \)
67 \( 1 + (0.629 + 0.777i)T \)
71 \( 1 + (-0.0672 - 0.997i)T \)
73 \( 1 + (-0.930 - 0.365i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.316 + 0.948i)T \)
97 \( 1 + (0.987 - 0.156i)T \)
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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

−17.693349626328562044844922775994, −16.9893687180507118431140117704, −16.79465152685246585967618783919, −15.816624963216867032911747834946, −14.89436352003943177442717155573, −14.353299529667399389000314739132, −13.571410982590957506266665776494, −12.95566358424644369400007726776, −12.55005826518611912762926864332, −11.70023355384616504409461025310, −11.24501231317597493955448915892, −10.63070922698480305307811838187, −9.57957256376594742328888339739, −9.05242794539194266004603434963, −8.800842681234365811190754478843, −7.929115501189192438361315576392, −6.93959759438614148001083865898, −6.00768067478216071421802840230, −5.38014782506410039578875994309, −4.51242685340588216659803769660, −3.94541784858375802501117666317, −3.48614985843036715095186463710, −2.14078260765510665554944088757, −1.68899066639066201117422635608, −0.93493378039579488620867036659, 0.257862770937665449220064668471, 1.45265561465490211691547449579, 2.61156364161395266519307826888, 3.34935192799397753634274829512, 4.08215718004836832566011995958, 4.76141883252119675848731564489, 5.78083249792726541851772937644, 6.19839478708483520457004067526, 6.95804767277514763580431283100, 7.47255299063503530432090806751, 8.125460097136784634141763065129, 9.037825638617062480903409620073, 9.5191372044051416349597371020, 10.38097649516828333343926516277, 10.93595306181361832738132317732, 11.827375026346895169454164186086, 12.599766810108717922901707747836, 13.24989453851333834377780226456, 13.97946210938150821689876100855, 14.6379758639801328751625936798, 14.99593862032688192455185246233, 15.54897053187784591267539677448, 16.47108620290041727156398191693, 16.95011999109160481944239196762, 17.6559187379338529119858901350

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