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Properties

Label 1-4563-4563.1055-r0-0-0
Degree $1$
Conductor $4563$
Sign $-0.993 - 0.109i$
Analytic cond. $21.1904$
Root an. cond. $21.1904$
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  + (−0.807 − 0.589i)2-s + (0.303 + 0.952i)4-s + (0.416 + 0.909i)5-s + (0.367 + 0.930i)7-s + (0.316 − 0.948i)8-s + (0.200 − 0.979i)10-s + (0.556 − 0.830i)11-s + (0.252 − 0.967i)14-s + (−0.815 + 0.579i)16-s + (−0.748 + 0.663i)17-s + (0.866 + 0.5i)19-s + (−0.739 + 0.673i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (−0.653 + 0.757i)25-s + ⋯
L(s)  = 1  + (−0.807 − 0.589i)2-s + (0.303 + 0.952i)4-s + (0.416 + 0.909i)5-s + (0.367 + 0.930i)7-s + (0.316 − 0.948i)8-s + (0.200 − 0.979i)10-s + (0.556 − 0.830i)11-s + (0.252 − 0.967i)14-s + (−0.815 + 0.579i)16-s + (−0.748 + 0.663i)17-s + (0.866 + 0.5i)19-s + (−0.739 + 0.673i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (−0.653 + 0.757i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $-0.993 - 0.109i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ -0.993 - 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009730169666 + 0.1764783894i\)
\(L(\frac12)\) \(\approx\) \(0.009730169666 + 0.1764783894i\)
\(L(1)\) \(\approx\) \(0.6917587033 + 0.04881060615i\)
\(L(1)\) \(\approx\) \(0.6917587033 + 0.04881060615i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.807 - 0.589i)T \)
5 \( 1 + (0.416 + 0.909i)T \)
7 \( 1 + (0.367 + 0.930i)T \)
11 \( 1 + (0.556 - 0.830i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.589 - 0.807i)T \)
31 \( 1 + (-0.944 - 0.329i)T \)
37 \( 1 + (-0.774 - 0.632i)T \)
41 \( 1 + (-0.0536 + 0.998i)T \)
43 \( 1 + (-0.329 - 0.944i)T \)
47 \( 1 + (-0.213 - 0.977i)T \)
53 \( 1 + (0.748 - 0.663i)T \)
59 \( 1 + (-0.579 + 0.815i)T \)
61 \( 1 + (0.476 + 0.879i)T \)
67 \( 1 + (-0.791 - 0.611i)T \)
71 \( 1 + (-0.992 + 0.120i)T \)
73 \( 1 + (-0.960 + 0.278i)T \)
79 \( 1 + (-0.673 - 0.739i)T \)
83 \( 1 + (-0.891 - 0.452i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.995 + 0.0938i)T \)
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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

−17.855493532205339988326102493408, −17.28976050284408651766924611789, −16.569076159653038990929295039255, −16.06813640118918747534661164132, −15.44731775753130423366789126818, −14.40088632262967166202574242948, −14.04119740548731870488886294888, −13.36121104030592847835918719878, −12.41664556185761357872063708170, −11.68372712381929221563402686934, −10.917151396461548858847596092450, −10.13605997729006566947170206916, −9.59025202166753272006958280451, −8.95522973609828083107513324951, −8.32854547005491392423972581179, −7.4353000073484758063310281636, −6.98389306110266378699995265174, −6.22612662731594057681486115772, −5.20118665747009952874712718606, −4.766353167638777899344683289805, −4.02894864286503333400932413735, −2.65153409349958025032929120326, −1.59496782537386456596637593272, −1.23943815560971045472599282824, −0.05995380601489259869177263194, 1.48206926972671315826376690476, 1.98475673882178401620181811217, 2.80551533097158949774611602423, 3.50319145080808020068360455970, 4.22765338004499299278731348770, 5.639075962391428155992358129411, 6.04058911660762358565956255617, 6.93728317045603971315950019155, 7.65895684772505525983282036976, 8.52719753708747400935036183403, 8.91878592894011730792870099807, 9.8306480080482049880815898424, 10.31758612946526699757694657232, 11.15378033388292729600219244957, 11.68007749572995301359579891731, 12.11235288771180650669638430515, 13.21546110767609341116424792762, 13.7509689507674352082603612897, 14.59985269689743923517107818999, 15.260761315393723845325232176837, 16.012479180914481609583822394612, 16.67590715827326526495531254581, 17.586430807643138962798034710253, 17.955906413044823490651549001691, 18.53877020155235861417139712363

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