Properties

Degree 1
Conductor 137
Sign $0.525 - 0.850i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.526 + 0.850i)2-s + (−0.403 + 0.914i)3-s + (−0.445 + 0.895i)4-s + (−0.228 − 0.973i)5-s + (−0.990 + 0.138i)6-s + (−0.183 + 0.982i)7-s + (−0.995 + 0.0922i)8-s + (−0.673 − 0.739i)9-s + (0.707 − 0.707i)10-s + (−0.895 − 0.445i)11-s + (−0.638 − 0.769i)12-s + (0.824 − 0.565i)13-s + (−0.932 + 0.361i)14-s + (0.982 + 0.183i)15-s + (−0.602 − 0.798i)16-s + (0.995 + 0.0922i)17-s + ⋯
L(s,χ)  = 1  + (0.526 + 0.850i)2-s + (−0.403 + 0.914i)3-s + (−0.445 + 0.895i)4-s + (−0.228 − 0.973i)5-s + (−0.990 + 0.138i)6-s + (−0.183 + 0.982i)7-s + (−0.995 + 0.0922i)8-s + (−0.673 − 0.739i)9-s + (0.707 − 0.707i)10-s + (−0.895 − 0.445i)11-s + (−0.638 − 0.769i)12-s + (0.824 − 0.565i)13-s + (−0.932 + 0.361i)14-s + (0.982 + 0.183i)15-s + (−0.602 − 0.798i)16-s + (0.995 + 0.0922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.525 - 0.850i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.525 - 0.850i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $0.525 - 0.850i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (21, \cdot )$
Sato-Tate  :  $\mu(136)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 137,\ (1:\ ),\ 0.525 - 0.850i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2229556741 - 0.1243122071i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2229556741 - 0.1243122071i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6649463317 + 0.4548600788i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6649463317 + 0.4548600788i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.6437693959089202159546069869, −27.760891481375339300279532178698, −26.42737608593658820044768345138, −25.48736280656016444069242876667, −23.68942770635730405432305289377, −23.44414565500057818247161482521, −22.6928931184752467792611291193, −21.45391765716290326753275695155, −20.30502563214403414868337720708, −19.24118439425510805941209655014, −18.5564148136352677284011432076, −17.693211600168043613060013829960, −16.16875330521990895029617411961, −14.60659225182594092926906058176, −13.7934839418311426295264254419, −12.89900845564586565632704125777, −11.78147401535450782515052316669, −10.79121950147671990797502880804, −10.10553075996020363513408866969, −8.05075308911121765775709733721, −6.85406361587031046878173354556, −5.85236143361293422918264541561, −4.21650995403292095425489269485, −2.910807593402460411419353922452, −1.54879896684749743579621729352, 0.08794184010087791158930989755, 3.09007715885011778656886785939, 4.36199290466013586164794300738, 5.46756966866661710228650321482, 6.04574872854155994654891224060, 8.19141734912780461792782325110, 8.71311755790148679916486720006, 10.11906593636527550650928188878, 11.75163054397880429611650928078, 12.55341485067600366798640867428, 13.72021198681046561183659833120, 15.309936856812338984522677732747, 15.67792031879246885699232296848, 16.56991109366596399939842236122, 17.53664129073187303780434411204, 18.784245634523018445264295252058, 20.63221393946642254680872137570, 21.211336217340078341745182189348, 22.11964073867083195266381757557, 23.273591677394364897326274685637, 23.89155405852976015742042656058, 25.20602165243358020100945833621, 25.87533737348163737843702021922, 27.13138288247420912750612754383, 27.992974974143044327953245688964

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