Properties

Degree 1
Conductor 173
Sign $-0.897 - 0.441i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.694 − 0.719i)2-s + (0.957 − 0.288i)3-s + (−0.0365 + 0.999i)4-s + (−0.997 + 0.0729i)5-s + (−0.872 − 0.489i)6-s + (−0.581 − 0.813i)7-s + (0.744 − 0.667i)8-s + (0.833 − 0.551i)9-s + (0.744 + 0.667i)10-s + (−0.934 + 0.357i)11-s + (0.252 + 0.967i)12-s + (−0.694 − 0.719i)13-s + (−0.181 + 0.983i)14-s + (−0.934 + 0.357i)15-s + (−0.997 − 0.0729i)16-s + (0.520 − 0.853i)17-s + ⋯
L(s,χ)  = 1  + (−0.694 − 0.719i)2-s + (0.957 − 0.288i)3-s + (−0.0365 + 0.999i)4-s + (−0.997 + 0.0729i)5-s + (−0.872 − 0.489i)6-s + (−0.581 − 0.813i)7-s + (0.744 − 0.667i)8-s + (0.833 − 0.551i)9-s + (0.744 + 0.667i)10-s + (−0.934 + 0.357i)11-s + (0.252 + 0.967i)12-s + (−0.694 − 0.719i)13-s + (−0.181 + 0.983i)14-s + (−0.934 + 0.357i)15-s + (−0.997 − 0.0729i)16-s + (0.520 − 0.853i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.897 - 0.441i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.897 - 0.441i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $-0.897 - 0.441i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (81, \cdot )$
Sato-Tate  :  $\mu(43)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ -0.897 - 0.441i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1393807562 - 0.5988517084i$
$L(\frac12,\chi)$  $\approx$  $0.1393807562 - 0.5988517084i$
$L(\chi,1)$  $\approx$  0.5700354287 - 0.4214951732i
$L(1,\chi)$  $\approx$  0.5700354287 - 0.4214951732i

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

−27.74355999920716131323047823186, −26.59105101075249452878032380575, −26.22381209172294956637293290204, −25.15968954120371234151388127658, −24.263582241245212921565515726146, −23.45815065510264498922051723461, −22.14179080763191365152080112444, −20.9246735279112487472397509049, −19.76827620210270816520096791085, −18.97804831539519142864487759792, −18.60484550446939980710952701425, −16.75887134324245280758911105903, −15.96596381098237294269726689392, −15.23487003962721145323599531945, −14.48287173112681859627969203103, −13.132336140957736106295865622555, −11.81121635564405555592272319657, −10.285997255942663666304957453302, −9.48347304991658590841076913080, −8.19685119831028023402410993514, −7.904229903000064907949024877578, −6.4017214487727650329336227304, −4.96963851403048158892702811425, −3.56187478462618401038547000584, −2.07435744554409336852843206646, 0.54213835914504275191786173599, 2.51112845652527268456792020765, 3.360413392802861415892632281957, 4.50122801701333012168147842121, 7.26906948626898988559024902509, 7.48663203616294204584533722398, 8.68370801870294646648476430302, 9.87883556877366264994494925621, 10.67335448097130326565410716973, 12.16432997928521290166115093911, 12.86611294181775308369859204166, 13.9308525367293952496723926586, 15.409758592799783261265587572116, 16.167595886275160989658735875404, 17.56390173813283602749374817718, 18.62554917473901746013613719429, 19.40303748594861900211331501805, 20.21107227016963385451541685577, 20.63496982945757805429892196178, 22.15153343351926957810555432163, 23.197505895096946261524896478614, 24.292338484631874555933513579925, 25.58181380539048299021710588405, 26.307066206588778101073462939987, 26.94371175708525950003351673120

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