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Properties

Label 1-137-137.47-r1-0-0
Degree $1$
Conductor $137$
Sign $-0.913 - 0.407i$
Analytic cond. $14.7226$
Root an. cond. $14.7226$
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  + (−0.798 + 0.602i)2-s + (0.638 + 0.769i)3-s + (0.273 − 0.961i)4-s + (−0.990 + 0.138i)5-s + (−0.973 − 0.228i)6-s + (0.673 − 0.739i)7-s + (0.361 + 0.932i)8-s + (−0.183 + 0.982i)9-s + (0.707 − 0.707i)10-s + (0.961 + 0.273i)11-s + (0.914 − 0.403i)12-s + (−0.998 + 0.0461i)13-s + (−0.0922 + 0.995i)14-s + (−0.739 − 0.673i)15-s + (−0.850 − 0.526i)16-s + (−0.361 + 0.932i)17-s + ⋯
L(s)  = 1  + (−0.798 + 0.602i)2-s + (0.638 + 0.769i)3-s + (0.273 − 0.961i)4-s + (−0.990 + 0.138i)5-s + (−0.973 − 0.228i)6-s + (0.673 − 0.739i)7-s + (0.361 + 0.932i)8-s + (−0.183 + 0.982i)9-s + (0.707 − 0.707i)10-s + (0.961 + 0.273i)11-s + (0.914 − 0.403i)12-s + (−0.998 + 0.0461i)13-s + (−0.0922 + 0.995i)14-s + (−0.739 − 0.673i)15-s + (−0.850 − 0.526i)16-s + (−0.361 + 0.932i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.913 - 0.407i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ -0.913 - 0.407i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1108952181 + 0.5206107718i\)
\(L(\frac12)\) \(\approx\) \(-0.1108952181 + 0.5206107718i\)
\(L(1)\) \(\approx\) \(0.5534265816 + 0.3803471656i\)
\(L(1)\) \(\approx\) \(0.5534265816 + 0.3803471656i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.798 + 0.602i)T \)
3 \( 1 + (0.638 + 0.769i)T \)
5 \( 1 + (-0.990 + 0.138i)T \)
7 \( 1 + (0.673 - 0.739i)T \)
11 \( 1 + (0.961 + 0.273i)T \)
13 \( 1 + (-0.998 + 0.0461i)T \)
17 \( 1 + (-0.361 + 0.932i)T \)
19 \( 1 + (-0.895 + 0.445i)T \)
23 \( 1 + (-0.973 + 0.228i)T \)
29 \( 1 + (-0.228 - 0.973i)T \)
31 \( 1 + (0.317 + 0.948i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.948 - 0.317i)T \)
47 \( 1 + (-0.565 - 0.824i)T \)
53 \( 1 + (-0.317 + 0.948i)T \)
59 \( 1 + (-0.982 - 0.183i)T \)
61 \( 1 + (-0.183 - 0.982i)T \)
67 \( 1 + (0.0461 + 0.998i)T \)
71 \( 1 + (0.873 + 0.486i)T \)
73 \( 1 + (0.739 - 0.673i)T \)
79 \( 1 + (-0.769 - 0.638i)T \)
83 \( 1 + (-0.403 + 0.914i)T \)
89 \( 1 + (0.138 + 0.990i)T \)
97 \( 1 + (0.486 + 0.873i)T \)
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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

−27.54156449797040967749749511945, −26.96276260331438160348700810557, −25.78453031179782924142794302584, −24.65833199398829316075219767791, −24.197695785691041683229184194457, −22.54096259215074838887110326666, −21.42459218127184262404778726492, −20.14691807769526601186287992935, −19.66517619076079612768485368166, −18.727901804376084839674720805134, −17.89668569350531532893469423, −16.73069601984200600949370305373, −15.35779255708651412315910856256, −14.36033103865639425820955847991, −12.78005350150276317050127953233, −11.93430561025822349565881816694, −11.305025595421715584992114382623, −9.431050423902441964961209328548, −8.58205998822593112017436579421, −7.758587149533863537115143127097, −6.70082388931361770860163099473, −4.40272165472433436325592211107, −2.98780614196285216171702524890, −1.81351914829472397792387651032, −0.24315629058745959929165595717, 1.85247088997187309635612762075, 3.88570327795513191712992298246, 4.7866945378512275069246390854, 6.71169823759373016636258427568, 7.87750261178688251402966787238, 8.51444826746075512895615464617, 9.89194189246926821910713997977, 10.71907300295032973855799130026, 11.87707549866224203708560679505, 13.97007578366820628373454935574, 14.82636113535904066357137438147, 15.39312733953787396156608584400, 16.73652447357909172123220100075, 17.287564681104581651506070970950, 18.958799283140305140476024290750, 19.78885038662880982853375999913, 20.28357276367835294973875183757, 21.79010989099470349478667929413, 23.05216765538600958950021053585, 24.05479579621435622015794056499, 24.96058881504593079626099586386, 26.16214930246790216668673671201, 26.86738461084795734192134673177, 27.48950768915146383034863747870, 28.18680724102235243857740993449

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