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Properties

Label 1-579-579.467-r1-0-0
Degree $1$
Conductor $579$
Sign $0.867 - 0.498i$
Analytic cond. $62.2221$
Root an. cond. $62.2221$
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  + 2-s + 4-s + i·5-s + 7-s + 8-s + i·10-s − i·11-s − i·13-s + 14-s + 16-s − i·17-s − i·19-s + i·20-s − i·22-s + 23-s + ⋯
L(s)  = 1  + 2-s + 4-s + i·5-s + 7-s + 8-s + i·10-s − i·11-s − i·13-s + 14-s + 16-s − i·17-s − i·19-s + i·20-s − i·22-s + 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.867 - 0.498i$
Analytic conductor: \(62.2221\)
Root analytic conductor: \(62.2221\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 579,\ (1:\ ),\ 0.867 - 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.761475334 - 1.270124966i\)
\(L(\frac12)\) \(\approx\) \(4.761475334 - 1.270124966i\)
\(L(1)\) \(\approx\) \(2.331642151 - 0.1347164390i\)
\(L(1)\) \(\approx\) \(2.331642151 - 0.1347164390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
193 \( 1 \)
good2 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
13 \( 1 + iT \)
17 \( 1 \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 \)
31 \( 1 + iT \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 + iT \)
79 \( 1 \)
83 \( 1 - iT \)
89 \( 1 + T \)
97 \( 1 \)
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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

−23.461044484235290772391215141878, −22.13495954883197273069736292443, −21.310683402552105759476388366378, −20.761331045488705695430929022045, −20.11417898478633273326925730676, −19.16414639907528869678777496763, −17.912821040635896301036115902525, −16.8628045050948557452578618829, −16.47382039351824214553936524758, −15.16056287096221608894475737841, −14.69514393845272600334752566894, −13.74614888343458557905527804714, −12.75647346929152797746441446872, −12.21195765338743661323986350041, −11.35430388816865764584657554634, −10.394416869399405328411785921304, −9.17476511239009858802447763070, −8.13704228001923967713414333913, −7.282121915476593176933136596649, −6.15127087126654601757322775926, −5.03811897083601764561453705229, −4.570175366159672133926958584101, −3.6059113466451659228803714040, −1.87309527548061833662403556149, −1.49934661224071277982050261247, 0.84457087179529068779103149310, 2.41747529911896613084185468123, 3.03358032935120419679761907308, 4.16074034235036965185145355450, 5.29071691036094500180250932243, 5.956358619186147139752744783092, 7.20973317013316825512437953883, 7.6935331201305847431955346175, 9.04172256567589830730187324472, 10.56739845302533882824167515224, 11.05318086526408330444931009482, 11.665453000303862415530885936603, 12.88864591979728886557384310525, 13.82329963416363560149238072063, 14.330142000470276270723093985186, 15.25932253290158639546160233706, 15.79767230518782239006727616659, 17.07122638134899019892097765586, 17.90115140248223378062829988860, 18.85907913730109064268254907078, 19.731521839290655035758335610949, 20.733908368679206715949469854902, 21.36871454434205118227506426480, 22.19755798128295701703091883502, 22.814768389820712610532900250382

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