Properties

Label 1-1512-1512.517-r1-0-0
Degree 11
Conductor 15121512
Sign 0.116+0.993i0.116 + 0.993i
Analytic cond. 162.486162.486
Root an. cond. 162.486162.486
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(1512s/2ΓR(s+1)L(s)=((0.116+0.993i)Λ(1s)\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}
Λ(s)=(1512s/2ΓR(s+1)L(s)=((0.116+0.993i)Λ(1s)\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: 11
Conductor: 15121512    =    233372^{3} \cdot 3^{3} \cdot 7
Sign: 0.116+0.993i0.116 + 0.993i
Analytic conductor: 162.486162.486
Root analytic conductor: 162.486162.486
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1512(517,)\chi_{1512} (517, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1512, (1: ), 0.116+0.993i)(1,\ 1512,\ (1:\ ),\ 0.116 + 0.993i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.9682481973+0.8616677314i0.9682481973 + 0.8616677314i
L(12)L(\frac12) \approx 0.9682481973+0.8616677314i0.9682481973 + 0.8616677314i
L(1)L(1) \approx 0.9386139308+0.02876477769i0.9386139308 + 0.02876477769i
L(1)L(1) \approx 0.9386139308+0.02876477769i0.9386139308 + 0.02876477769i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
7 1 1
good5 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
11 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
13 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
17 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
23 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
29 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
31 1+(0.1730.984i)T 1 + (-0.173 - 0.984i)T
37 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
41 1+(0.766+0.642i)T 1 + (-0.766 + 0.642i)T
43 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
47 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
53 1T 1 - T
59 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
61 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
67 1+(0.766+0.642i)T 1 + (-0.766 + 0.642i)T
71 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
73 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
79 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
83 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
89 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
97 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line