Properties

Degree 1
Conductor 173
Sign $-0.299 + 0.954i$
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

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.299 + 0.954i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.299 + 0.954i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $-0.299 + 0.954i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (92, \cdot )$
Sato-Tate  :  $\mu(86)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 173,\ (0:\ ),\ -0.299 + 0.954i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8540743245 + 1.163058073i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8540743245 + 1.163058073i\)
\(L(\chi,1)\)  \(\approx\)  \(1.074238565 + 0.7759639016i\)
\(L(1,\chi)\)  \(\approx\)  \(1.074238565 + 0.7759639016i\)

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

−27.57678184129348296634688218726, −26.430621410769935346764388003855, −24.67750373946769186106937402689, −24.29396826669117440506271659310, −23.165934742321005244586797372755, −22.21583474567681972988528574421, −21.700099690737450628983409641078, −20.55971091466876300186854391889, −19.58801752298381789678307288183, −18.29708147816718370143801000206, −17.50807761685822315089794048943, −16.64938462535941894796830139285, −15.04828304450475008148203241077, −13.830259179936580459389277661482, −13.3936341403782862042297314393, −11.92247991060861092597864863266, −11.3771758336219808157351437586, −10.17560202568824084344406840871, −9.42077119637239372304053695631, −7.11399882584394798054074360171, −6.33692171829863130885483079812, −5.02247861726583872257081876975, −4.305161280115565000550929780206, −2.26558195496639146590604753245, −1.20956109695853529432662497208, 1.99310638899563247917122386012, 3.84118411635494647592514672859, 5.17878260026623028223426807257, 5.80019474226673481089339094419, 6.67421205405713481962073826652, 8.26723835303681227065857604374, 9.4547043921702638820133576391, 10.78404209101740260285149770279, 12.08312498775940046267169914937, 12.64310494356244487092951136212, 14.09018329289259623305130188063, 14.894993771810305280645861774099, 15.96144247182325625543362735304, 17.1225639002883875635479629303, 17.50120337475114156436862093333, 18.54125160273255520359941671176, 20.495301059394595091517754879802, 21.53552759170434421008225084492, 22.05920249216389526334593266316, 22.70742501421540470901179708185, 24.27223635109739072550945304083, 24.52423687365407675706490153653, 25.58994840485868174129156657624, 26.83123909354831349339945924051, 27.69588335121419913031094438665

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