Properties

Degree 1
Conductor 137
Sign $0.924 - 0.380i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.739 + 0.673i)2-s + (0.526 − 0.850i)3-s + (0.0922 − 0.995i)4-s + (0.673 − 0.739i)5-s + (0.183 + 0.982i)6-s + (0.273 + 0.961i)7-s + (0.602 + 0.798i)8-s + (−0.445 − 0.895i)9-s + i·10-s + (−0.0922 + 0.995i)11-s + (−0.798 − 0.602i)12-s + (0.961 − 0.273i)13-s + (−0.850 − 0.526i)14-s + (−0.273 − 0.961i)15-s + (−0.982 − 0.183i)16-s + (0.602 − 0.798i)17-s + ⋯
L(s,χ)  = 1  + (−0.739 + 0.673i)2-s + (0.526 − 0.850i)3-s + (0.0922 − 0.995i)4-s + (0.673 − 0.739i)5-s + (0.183 + 0.982i)6-s + (0.273 + 0.961i)7-s + (0.602 + 0.798i)8-s + (−0.445 − 0.895i)9-s + i·10-s + (−0.0922 + 0.995i)11-s + (−0.798 − 0.602i)12-s + (0.961 − 0.273i)13-s + (−0.850 − 0.526i)14-s + (−0.273 − 0.961i)15-s + (−0.982 − 0.183i)16-s + (0.602 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.924 - 0.380i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.924 - 0.380i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $0.924 - 0.380i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (118, \cdot )$
Sato-Tate  :  $\mu(68)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 137,\ (0:\ ),\ 0.924 - 0.380i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9998934983 - 0.1975461597i$
$L(\frac12,\chi)$  $\approx$  $0.9998934983 - 0.1975461597i$
$L(\chi,1)$  $\approx$  0.9817378420 - 0.07536045959i
$L(1,\chi)$  $\approx$  0.9817378420 - 0.07536045959i

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

−28.38862956534548518950774906853, −27.43831199830491234186307287628, −26.527968040372361046441455174464, −26.03892387762117692270027304651, −25.145024914924934375588775509150, −23.38283748227581264984983946462, −22.116694931344569443573225578539, −21.18032741811745007864779300222, −20.813385931956955604615519470573, −19.39211327435526453558257901093, −18.75315005901913197266989488291, −17.27631119303580193783937908057, −16.69548667184931011805396637239, −15.34114421581632623547625658550, −13.94655097909974141415984477404, −13.36342055940206829288157342622, −11.29713888717167285819875499068, −10.65020168873722867013437797951, −9.88820770656453033158046253888, −8.66960637868393747238289030966, −7.68735808629197046003242095562, −6.05597736010578816749857736009, −4.03133775925398762819939088450, −3.23786421257657027701816659306, −1.72981056859313001718990935433, 1.33748372354046007387347848263, 2.41229882200272799739454102682, 4.989977055160969719212714694152, 6.072085832461558748507177001813, 7.18455849934275084766310737328, 8.62894419021820084364536154078, 8.88013021917422276443341172353, 10.29245854293738972637927640697, 12.01803246374807629903696984207, 13.026391403074239099296281233638, 14.208607078132138411931093885057, 15.12955628308284920401779225420, 16.29083056206016544630559326431, 17.63482556645223657988397471944, 18.11434200782528408638995581142, 19.07570627561382751970486331077, 20.315121805572186301782369863883, 21.00008545745091955143462218561, 22.84871662070236777430987502759, 23.86966420404640354292613117800, 24.80355221680464717605863266647, 25.400742807950294882689405363214, 25.935312351393173834266119971465, 27.61018033937361342202205741204, 28.32078737178788797770250702618

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