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Properties

Label 1-1512-1512.517-r1-0-0
Degree $1$
Conductor $1512$
Sign $0.116 + 0.993i$
Analytic cond. $162.486$
Root an. cond. $162.486$
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  + (−0.939 − 0.342i)5-s + (0.939 − 0.342i)11-s + (0.766 − 0.642i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)29-s + (−0.173 − 0.984i)31-s + (0.5 + 0.866i)37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 + 0.984i)47-s − 53-s − 55-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (0.939 − 0.342i)11-s + (0.766 − 0.642i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)29-s + (−0.173 − 0.984i)31-s + (0.5 + 0.866i)37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 + 0.984i)47-s − 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(162.486\)
Root analytic conductor: \(162.486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1512,\ (1:\ ),\ 0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9682481973 + 0.8616677314i\)
\(L(\frac12)\) \(\approx\) \(0.9682481973 + 0.8616677314i\)
\(L(1)\) \(\approx\) \(0.9386139308 + 0.02876477769i\)
\(L(1)\) \(\approx\) \(0.9386139308 + 0.02876477769i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−20.13845070932247019675766864074, −19.53390578420467860852291960394, −18.74612089840363804314395778276, −18.20229039541939547784660860413, −17.19463271401509668753935421769, −16.333129471298763641269019694601, −15.88438008506985446249523534699, −14.80808389266050556057350294321, −14.43883696430113239690479110872, −13.48160901024381398916124446518, −12.456298749948777664499323679620, −11.86236517526837462669802275851, −11.09909101844126560993240958409, −10.499253549726711816702562792496, −9.16470211467158719887459822650, −8.8449109406728435280423595892, −7.69930328883675126700291220892, −6.93840401333473737992810435167, −6.40677583887943168595835802912, −5.094190492805490187336681501791, −4.2466212434148353722001468451, −3.56823882758145740415582309353, −2.5886703146960842256255394864, −1.37717685740516128047297166038, −0.30177908935094189634151479084, 0.915081681858848643349225585844, 1.7092633805063098317785643579, 3.32819058294500559537633800554, 3.73334130035465721472195483940, 4.62172597723626895692250800704, 5.83030844217871552936217539432, 6.33670466162845926507169542724, 7.718270210286064293888776700579, 8.01551580184058831961156746637, 8.953997958663959201488116031540, 9.760131627440575898419212261696, 10.86361373622374790851331000759, 11.405582693616021140733351542185, 12.22840983026363607194811667643, 12.91804084474078961192675427991, 13.73308309359414041176064055215, 14.81157469850015209838050101600, 15.221627504010549677080033210806, 16.126111844590178211853958181909, 16.86666203865229692343102397913, 17.38016561498520009201261991412, 18.6598368681721943681760672716, 19.05221568739165581612825640143, 19.825524534783789984681950498276, 20.562211042751507008008439590499

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