Properties

Degree 1
Conductor 173
Sign $0.956 + 0.292i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.989 − 0.145i)2-s + (−0.694 + 0.719i)3-s + (0.957 − 0.288i)4-s + (0.833 + 0.551i)5-s + (−0.581 + 0.813i)6-s + (0.252 − 0.967i)7-s + (0.905 − 0.424i)8-s + (−0.0365 − 0.999i)9-s + (0.905 + 0.424i)10-s + (−0.976 + 0.217i)11-s + (−0.457 + 0.889i)12-s + (0.989 − 0.145i)13-s + (0.109 − 0.994i)14-s + (−0.976 + 0.217i)15-s + (0.833 − 0.551i)16-s + (−0.322 + 0.946i)17-s + ⋯
L(s,χ)  = 1  + (0.989 − 0.145i)2-s + (−0.694 + 0.719i)3-s + (0.957 − 0.288i)4-s + (0.833 + 0.551i)5-s + (−0.581 + 0.813i)6-s + (0.252 − 0.967i)7-s + (0.905 − 0.424i)8-s + (−0.0365 − 0.999i)9-s + (0.905 + 0.424i)10-s + (−0.976 + 0.217i)11-s + (−0.457 + 0.889i)12-s + (0.989 − 0.145i)13-s + (0.109 − 0.994i)14-s + (−0.976 + 0.217i)15-s + (0.833 − 0.551i)16-s + (−0.322 + 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.956 + 0.292i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.956 + 0.292i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $0.956 + 0.292i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (119, \cdot )$
Sato-Tate  :  $\mu(43)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ 0.956 + 0.292i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.876847448 + 0.2802295426i$
$L(\frac12,\chi)$  $\approx$  $1.876847448 + 0.2802295426i$
$L(\chi,1)$  $\approx$  1.677990241 + 0.1683134542i
$L(1,\chi)$  $\approx$  1.677990241 + 0.1683134542i

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

−27.97958827052011549440623956492, −25.93943995118423596628501311824, −25.24152384581093818418390284628, −24.29303058948277389300206780653, −23.81538829529494490345792982483, −22.62200475721369706219622051999, −21.72475740879044737493604639618, −21.05140035644040996367143091133, −19.871212706992578484345450646385, −18.29462131690669753258340722477, −17.821464635465934369420598849510, −16.30808466178222244252739850674, −15.84093280289030566819530504285, −14.21273786984288514754390079978, −13.30205218186758767401083084809, −12.70015462958501450103844601710, −11.59817700139675737217418900207, −10.74584860809814908230354201, −8.948900766153493822592721380511, −7.66949679994993961632178722426, −6.33748041000665972817101994249, −5.5405181288302253767714943356, −4.84514669999897090429544552666, −2.71920138043768147462641438782, −1.67928970821861501810628711282, 1.74891615499900736021875583065, 3.440486282155166816937083883842, 4.380480230331056932863850276339, 5.71507094864703534488179533481, 6.30632026392288331719608776085, 7.75877326582451667050311205629, 9.89509399726948612739996505071, 10.57910958883835362056265730274, 11.20340286106969550888578303113, 12.69606339165034835845579651778, 13.619941290281778659474854025433, 14.59797757417207842186763892988, 15.57351750945917665102653711764, 16.5883500308008739681231440897, 17.54540727085980385454386652240, 18.66857798221663202048807815251, 20.47906990052443160011654175271, 20.831423810298510957700660624679, 21.82259842584405919727436859719, 22.706769537094695446813054786056, 23.39444712360303782639931334269, 24.27699708052307102100733890816, 25.882998275539948481697385492665, 26.24419171693598432927674687257, 27.76301255290577542101576901792

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