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Properties

Label 1-1148-1148.551-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.924 + 0.380i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  + (−0.5 + 0.866i)3-s + (0.978 − 0.207i)5-s + (−0.5 − 0.866i)9-s + (0.978 + 0.207i)11-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.978 + 0.207i)17-s + (−0.104 + 0.994i)19-s + (−0.913 − 0.406i)23-s + (0.913 − 0.406i)25-s + 27-s + (0.309 − 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.669 + 0.743i)33-s + (−0.978 + 0.207i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.978 − 0.207i)5-s + (−0.5 − 0.866i)9-s + (0.978 + 0.207i)11-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.978 + 0.207i)17-s + (−0.104 + 0.994i)19-s + (−0.913 − 0.406i)23-s + (0.913 − 0.406i)25-s + 27-s + (0.309 − 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.669 + 0.743i)33-s + (−0.978 + 0.207i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.924 + 0.380i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.924 + 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.696021885 + 0.3356100546i\)
\(L(\frac12)\) \(\approx\) \(1.696021885 + 0.3356100546i\)
\(L(1)\) \(\approx\) \(1.181686663 + 0.2059558842i\)
\(L(1)\) \(\approx\) \(1.181686663 + 0.2059558842i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.978 - 0.207i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.104 - 0.994i)T \)
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

−21.48046960454629384613238335390, −20.45028567488276660015976600329, −19.512321738646754457082929127819, −18.8793649104580295416841476448, −17.97550957137488786538454728156, −17.6540059794721013544336284800, −16.63390590593339785977403113777, −16.21474962073913213982068463898, −14.74781732342322123495075894831, −13.97997872180979525979066270350, −13.5926848931604604683728240201, −12.6162526858031765149837410030, −11.848667695179201979117557521787, −11.083164274111598507522195663, −10.29404649220268295107738228930, −9.19352604460000515851009360781, −8.59978259327635374676662720558, −7.2738316823230150098117768643, −6.74342739014370578434764158447, −5.87779643123406321435530359448, −5.32247663446288116880308505799, −3.97645878082167632129119818260, −2.81505867701141539543690946297, −1.74348959365163017587269784151, −1.09454848470100862674934588480, 0.95282229559490033338565932117, 2.01679834849401565978991952686, 3.43157088522392359912753854541, 4.05974873853321588982759899125, 5.19052946191423549921237002570, 5.96333492699827185379890192003, 6.36665770261051659420502523780, 7.84177989135108950616012429063, 8.84584129155693575533102624582, 9.535303637807360510067259902576, 10.28489710535484297101574766968, 10.82059243807267620816849458924, 12.06901427208038921460877741885, 12.45051520745903634173437747356, 13.76639729580064656216989200960, 14.320820406830280267047099732448, 15.145333104280356792583810965206, 16.08283851240299672716721099169, 16.78636296849486385758794456203, 17.310433306905244365466803437352, 18.08320155457899027626378728446, 18.90571876987903033055517367191, 20.164841210280773198010655991117, 20.70632349973110454163532810941, 21.29346933959300765719919382637

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