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Properties

Label 1-6027-6027.2483-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.984 - 0.174i$
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  + (−0.193 + 0.981i)2-s + (−0.925 − 0.379i)4-s + (−0.772 − 0.635i)5-s + (0.550 − 0.834i)8-s + (0.772 − 0.635i)10-s + (0.0149 + 0.999i)11-s + (0.753 + 0.657i)13-s + (0.712 + 0.701i)16-s + (−0.791 − 0.611i)17-s + (−0.104 + 0.994i)19-s + (0.473 + 0.880i)20-s + (−0.983 − 0.178i)22-s + (0.999 − 0.0299i)23-s + (0.193 + 0.981i)25-s + (−0.791 + 0.611i)26-s + ⋯
L(s)  = 1  + (−0.193 + 0.981i)2-s + (−0.925 − 0.379i)4-s + (−0.772 − 0.635i)5-s + (0.550 − 0.834i)8-s + (0.772 − 0.635i)10-s + (0.0149 + 0.999i)11-s + (0.753 + 0.657i)13-s + (0.712 + 0.701i)16-s + (−0.791 − 0.611i)17-s + (−0.104 + 0.994i)19-s + (0.473 + 0.880i)20-s + (−0.983 − 0.178i)22-s + (0.999 − 0.0299i)23-s + (0.193 + 0.981i)25-s + (−0.791 + 0.611i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.984 - 0.174i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.984 - 0.174i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06477463927 + 0.7372569114i\)
\(L(\frac12)\) \(\approx\) \(-0.06477463927 + 0.7372569114i\)
\(L(1)\) \(\approx\) \(0.6276291991 + 0.3927880578i\)
\(L(1)\) \(\approx\) \(0.6276291991 + 0.3927880578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.193 + 0.981i)T \)
5 \( 1 + (-0.772 - 0.635i)T \)
11 \( 1 + (0.0149 + 0.999i)T \)
13 \( 1 + (0.753 + 0.657i)T \)
17 \( 1 + (-0.791 - 0.611i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.999 - 0.0299i)T \)
29 \( 1 + (-0.691 + 0.722i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.971 + 0.237i)T \)
43 \( 1 + (-0.0448 + 0.998i)T \)
47 \( 1 + (-0.193 + 0.981i)T \)
53 \( 1 + (-0.925 - 0.379i)T \)
59 \( 1 + (-0.842 - 0.538i)T \)
61 \( 1 + (-0.525 + 0.850i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (-0.691 - 0.722i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.946 + 0.323i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−17.47644437065748993616260885882, −16.936087515213772425022116224449, −15.99422362878193829090966103770, −15.3308328686638858561926118030, −14.81579171800441581792805069583, −13.81151642278964210185323639357, −13.316898203718352017589447195087, −12.832191521115901963264313111403, −11.74146688810186473664866538418, −11.44026948393232038512486092964, −10.64311803522087856931713189520, −10.556164759730467047198043973984, −9.29389295180715922114220758904, −8.76835759473496757890095207335, −8.12095123701213666880742654188, −7.52609404564534696596156524395, −6.525859536563516807989192033317, −5.852782595978599065304496260538, −4.8147227756194397592715549961, −4.14326507272097247534091389869, −3.40419465256315928363120983384, −2.925958822553926866956773530613, −2.13941214861905409798923222167, −0.97442173778417762930286958515, −0.267878885911360394402091310296, 1.06170060832771274484117176808, 1.6790854180198363630277902127, 3.066045649461583170235431164339, 3.97628356812968772519846039410, 4.62138195679718241540218591396, 4.95398329685433380629137524553, 6.058177601192912026484353066182, 6.64814553089559137302782967634, 7.39258365176801099787087948521, 7.93155792083352170028040338839, 8.6496733372793180185419886627, 9.2747589335096245095657988425, 9.73566772101015458420872359096, 10.831918265403246340188222117948, 11.39232795732875217290237451499, 12.36659158902629473370602198567, 12.82807800058438498291678889026, 13.53320440299085573650783872707, 14.27684925081642312125751115382, 15.01364173052716343112369410733, 15.45356504573087816139110081070, 16.17397489136645967681759366475, 16.600580515815918817807777640914, 17.2236940406254497647868753163, 18.05389792781493994862016688928

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