Properties

Degree 1
Conductor 173
Sign $0.469 + 0.883i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.252 − 0.967i)2-s + (0.581 + 0.813i)3-s + (−0.872 + 0.489i)4-s + (−0.520 − 0.853i)5-s + (0.639 − 0.768i)6-s + (−0.744 + 0.667i)7-s + (0.694 + 0.719i)8-s + (−0.322 + 0.946i)9-s + (−0.694 + 0.719i)10-s + (−0.391 + 0.920i)11-s + (−0.905 − 0.424i)12-s + (0.252 + 0.967i)13-s + (0.833 + 0.551i)14-s + (0.391 − 0.920i)15-s + (0.520 − 0.853i)16-s + (0.181 − 0.983i)17-s + ⋯
L(s,χ)  = 1  + (−0.252 − 0.967i)2-s + (0.581 + 0.813i)3-s + (−0.872 + 0.489i)4-s + (−0.520 − 0.853i)5-s + (0.639 − 0.768i)6-s + (−0.744 + 0.667i)7-s + (0.694 + 0.719i)8-s + (−0.322 + 0.946i)9-s + (−0.694 + 0.719i)10-s + (−0.391 + 0.920i)11-s + (−0.905 − 0.424i)12-s + (0.252 + 0.967i)13-s + (0.833 + 0.551i)14-s + (0.391 − 0.920i)15-s + (0.520 − 0.853i)16-s + (0.181 − 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.469 + 0.883i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.469 + 0.883i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $0.469 + 0.883i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (21, \cdot )$
Sato-Tate  :  $\mu(86)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ 0.469 + 0.883i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6068831940 + 0.3647952059i$
$L(\frac12,\chi)$  $\approx$  $0.6068831940 + 0.3647952059i$
$L(\chi,1)$  $\approx$  0.7866158121 + 0.03160428444i
$L(1,\chi)$  $\approx$  0.7866158121 + 0.03160428444i

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

−26.76768813923210618914444302313, −26.299340730910906198499089740500, −25.545370879805178288581090737141, −24.50738137198485629994936081310, −23.42944042871957193298839148599, −23.09816908240764987166521364248, −21.8059459739385509631533381482, −20.05504492224921207636661792731, −19.21738554247221473103582195238, −18.64287057458267273094375931368, −17.59403463909889715216836380093, −16.48149050759660657440637054275, −15.286065176845284522891568072305, −14.621634431407571931750512156617, −13.42382976340858240738420714845, −12.87242562197746532470357367483, −10.98054444558417030298614435404, −9.97348912986660039969326533645, −8.421338465438280231296017275357, −7.85503452719894185512684637188, −6.67544319494395062694746529040, −6.08057351757118513442401946852, −3.98727565599495992629150746436, −2.891823760768722656745611845985, −0.58793608725657310072772741516, 1.928572037667893249334677604317, 3.22287301906273530837508729839, 4.30505165277231628300588863754, 5.2278246543503876031256181788, 7.51334848992255409827658597502, 8.84856572816748613253153291797, 9.2528483718300921567148304368, 10.3159415650877774080504247506, 11.629658331796562556752400804641, 12.52609004140639597723372217653, 13.48511346056595917119037942137, 14.79777102441241041411802142421, 16.01833402938576509990086036898, 16.63938038110778912143593262025, 18.14848959425617264375275035548, 19.32600185561370415490391992339, 19.84302564996600401252560202910, 20.9513226626505554084423850174, 21.40326260269645765690953094825, 22.65174449653357251265781970151, 23.4727137528153690289844034780, 25.25761790090129859952104613227, 25.774527944845447352739375877577, 27.03893093317763469549632339008, 27.68890908210531040960985148693

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