Properties

Degree 1
Conductor 173
Sign $0.839 - 0.543i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.934 − 0.357i)2-s + (0.905 + 0.424i)3-s + (0.744 + 0.667i)4-s + (0.109 − 0.994i)5-s + (−0.694 − 0.719i)6-s + (0.989 + 0.145i)7-s + (−0.457 − 0.889i)8-s + (0.639 + 0.768i)9-s + (−0.457 + 0.889i)10-s + (0.520 − 0.853i)11-s + (0.391 + 0.920i)12-s + (−0.934 − 0.357i)13-s + (−0.872 − 0.489i)14-s + (0.520 − 0.853i)15-s + (0.109 + 0.994i)16-s + (−0.0365 − 0.999i)17-s + ⋯
L(s,χ)  = 1  + (−0.934 − 0.357i)2-s + (0.905 + 0.424i)3-s + (0.744 + 0.667i)4-s + (0.109 − 0.994i)5-s + (−0.694 − 0.719i)6-s + (0.989 + 0.145i)7-s + (−0.457 − 0.889i)8-s + (0.639 + 0.768i)9-s + (−0.457 + 0.889i)10-s + (0.520 − 0.853i)11-s + (0.391 + 0.920i)12-s + (−0.934 − 0.357i)13-s + (−0.872 − 0.489i)14-s + (0.520 − 0.853i)15-s + (0.109 + 0.994i)16-s + (−0.0365 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.839 - 0.543i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.839 - 0.543i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $0.839 - 0.543i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (14, \cdot )$
Sato-Tate  :  $\mu(43)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ 0.839 - 0.543i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.073765854 - 0.3169267141i$
$L(\frac12,\chi)$  $\approx$  $1.073765854 - 0.3169267141i$
$L(\chi,1)$  $\approx$  1.014736750 - 0.1821929477i
$L(1,\chi)$  $\approx$  1.014736750 - 0.1821929477i

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

−27.27184181913310885295760224156, −26.45448696337928225092901639048, −25.883981825966639635625281358861, −24.8074328084943064802645030565, −24.17976780699505237343214761075, −23.0366391508836304187561995737, −21.47552653299846581614062697034, −20.56244348547137322645254635634, −19.3780704008136632093532066608, −18.99746638922109253851174583630, −17.614650454764159682430752012268, −17.36906867056671087717969222886, −15.429394992076656428198031668280, −14.64516936872778749305619842250, −14.304342627366265275516248073747, −12.5116088479564117830199437913, −11.21422360591992027070500532558, −10.18662750175946659331742785234, −9.153274707181696446171319834752, −8.02555632224962470382321994362, −7.19931835105016412221482842781, −6.37853405421362535722027206857, −4.39069140132646770241336598694, −2.52266701678701534662336658654, −1.73173807851816686335261864840, 1.33363151263977587944482434726, 2.55438092630868643902372849205, 4.01042916634488830497586997865, 5.31255615614524911805439422126, 7.34557542149862954222867268998, 8.32179253128400301539602950585, 8.96621708485764665238152858062, 9.89418849351226493842487376762, 11.15464152434361218317401812802, 12.20056691564425581480280580307, 13.42712381553309675361997465465, 14.6380627741555043386314901144, 15.730819069429031206071970593860, 16.73928726808991366899813896676, 17.50883396019418771831377002503, 18.881585827739168094567331198, 19.65438800203945498660040822367, 20.634959382558388073239265001895, 21.12143365054414405245575489652, 22.05369560920246963697979068226, 24.173714053627988380876648941475, 24.75295073288096623296157438781, 25.44136946171893387065487352294, 26.77484915287163810598076850754, 27.44800987920679798100109308662

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