Properties

Label 1-137-137.21-r1-0-0
Degree $1$
Conductor $137$
Sign $0.525 - 0.850i$
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  + (0.526 + 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

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

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: $\chi_{137} (21, \cdot )$
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

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.526 + 0.850i)T \)
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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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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