Dirichlet series
L(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) = 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
Invariants
Degree: | \(1\) |
Conductor: | \(137\) |
Sign: | $0.525 + 0.850i$ |
Analytic conductor: | \(14.7226\) |
Root analytic conductor: | \(14.7226\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{137} (124, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 137,\ (1:\ ),\ 0.525 + 0.850i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.2229556741 + 0.1243122071i\) |
\(L(\frac12)\) | \(\approx\) | \(0.2229556741 + 0.1243122071i\) |
\(L(1)\) | \(\approx\) | \(0.6649463317 - 0.4548600788i\) |
\(L(1)\) | \(\approx\) | \(0.6649463317 - 0.4548600788i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (0.526 - 0.850i)T \) |
3 | \( 1 + (-0.403 - 0.914i)T \) | |
5 | \( 1 + (-0.228 + 0.973i)T \) | |
7 | \( 1 + (-0.183 - 0.982i)T \) | |
11 | \( 1 + (-0.895 + 0.445i)T \) | |
13 | \( 1 + (0.824 + 0.565i)T \) | |
17 | \( 1 + (0.995 - 0.0922i)T \) | |
19 | \( 1 + (-0.961 + 0.273i)T \) | |
23 | \( 1 + (-0.990 + 0.138i)T \) | |
29 | \( 1 + (0.138 + 0.990i)T \) | |
31 | \( 1 + (0.873 - 0.486i)T \) | |
37 | \( 1 - iT \) | |
41 | \( 1 + (-0.707 + 0.707i)T \) | |
43 | \( 1 + (-0.486 + 0.873i)T \) | |
47 | \( 1 + (0.998 - 0.0461i)T \) | |
53 | \( 1 + (-0.873 - 0.486i)T \) | |
59 | \( 1 + (0.739 + 0.673i)T \) | |
61 | \( 1 + (-0.673 - 0.739i)T \) | |
67 | \( 1 + (-0.565 + 0.824i)T \) | |
71 | \( 1 + (-0.948 + 0.317i)T \) | |
73 | \( 1 + (-0.982 - 0.183i)T \) | |
79 | \( 1 + (-0.914 - 0.403i)T \) | |
83 | \( 1 + (-0.769 + 0.638i)T \) | |
89 | \( 1 + (-0.973 - 0.228i)T \) | |
97 | \( 1 + (-0.317 + 0.948i)T \) | |
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Imaginary part of the first few zeros on the critical line
−27.992974974143044327953245688964, −27.13138288247420912750612754383, −25.87533737348163737843702021922, −25.20602165243358020100945833621, −23.89155405852976015742042656058, −23.273591677394364897326274685637, −22.11964073867083195266381757557, −21.211336217340078341745182189348, −20.63221393946642254680872137570, −18.784245634523018445264295252058, −17.53664129073187303780434411204, −16.56991109366596399939842236122, −15.67792031879246885699232296848, −15.309936856812338984522677732747, −13.72021198681046561183659833120, −12.55341485067600366798640867428, −11.75163054397880429611650928078, −10.11906593636527550650928188878, −8.71311755790148679916486720006, −8.19141734912780461792782325110, −6.04574872854155994654891224060, −5.46756966866661710228650321482, −4.36199290466013586164794300738, −3.09007715885011778656886785939, −0.08794184010087791158930989755, 1.54879896684749743579621729352, 2.910807593402460411419353922452, 4.21650995403292095425489269485, 5.85236143361293422918264541561, 6.85406361587031046878173354556, 8.05075308911121765775709733721, 10.10553075996020363513408866969, 10.79121950147671990797502880804, 11.78147401535450782515052316669, 12.89900845564586565632704125777, 13.7934839418311426295264254419, 14.60659225182594092926906058176, 16.16875330521990895029617411961, 17.693211600168043613060013829960, 18.5564148136352677284011432076, 19.24118439425510805941209655014, 20.30502563214403414868337720708, 21.45391765716290326753275695155, 22.6928931184752467792611291193, 23.44414565500057818247161482521, 23.68942770635730405432305289377, 25.48736280656016444069242876667, 26.42737608593658820044768345138, 27.760891481375339300279532178698, 28.6437693959089202159546069869