Properties

Degree 1
Conductor 173
Sign $0.905 + 0.425i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.791 + 0.611i)2-s + (0.457 − 0.889i)3-s + (0.252 + 0.967i)4-s + (0.872 + 0.489i)5-s + (0.905 − 0.424i)6-s + (0.934 − 0.357i)7-s + (−0.391 + 0.920i)8-s + (−0.581 − 0.813i)9-s + (0.391 + 0.920i)10-s + (−0.833 + 0.551i)11-s + (0.976 + 0.217i)12-s + (−0.791 − 0.611i)13-s + (0.957 + 0.288i)14-s + (0.833 − 0.551i)15-s + (−0.872 + 0.489i)16-s + (−0.639 − 0.768i)17-s + ⋯
L(s,χ)  = 1  + (0.791 + 0.611i)2-s + (0.457 − 0.889i)3-s + (0.252 + 0.967i)4-s + (0.872 + 0.489i)5-s + (0.905 − 0.424i)6-s + (0.934 − 0.357i)7-s + (−0.391 + 0.920i)8-s + (−0.581 − 0.813i)9-s + (0.391 + 0.920i)10-s + (−0.833 + 0.551i)11-s + (0.976 + 0.217i)12-s + (−0.791 − 0.611i)13-s + (0.957 + 0.288i)14-s + (0.833 − 0.551i)15-s + (−0.872 + 0.489i)16-s + (−0.639 − 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.905 + 0.425i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.905 + 0.425i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $0.905 + 0.425i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (49, \cdot )$
Sato-Tate  :  $\mu(86)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 173,\ (0:\ ),\ 0.905 + 0.425i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.121641230 + 0.4735989040i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.121641230 + 0.4735989040i\)
\(L(\chi,1)\)  \(\approx\)  \(1.874022707 + 0.3236198238i\)
\(L(1,\chi)\)  \(\approx\)  \(1.874022707 + 0.3236198238i\)

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

−27.565765597896438305613318790306, −26.6022407867904918458060322497, −25.329612432933420307490598015957, −24.44822196574567576976200354307, −23.615828794669575323438470476954, −22.073652364055732254031274604811, −21.37832390439732643038374757423, −21.11086989011703348510736086416, −19.99205161737305779522312408535, −18.97145132661310471893521032311, −17.58740755093326374785245531259, −16.42128426383714134755047767412, −15.20424212112151469639986504464, −14.48009397590750147458123173110, −13.56861416326682796181846396833, −12.569993012768463631267387907520, −11.129736019855378601939506500907, −10.4374827704622450336292576724, −9.269711257222317411850540049285, −8.34949657941427168932841341861, −6.251489716332019610791608254575, −4.98207574666722954453781965912, −4.56124132430803622280128512826, −2.78848556309286202624984777449, −1.90858283590969110487136056504, 2.04140916820326217871728423699, 2.88052089550982103108840283111, 4.69580935773726750227730256717, 5.771970355163241518990760600931, 7.06429635911720400035012208828, 7.62744862966540321873235674155, 8.8815244423324934149966128237, 10.52426405922100176476456461147, 11.818016747495404244894469780879, 13.03564159224699400771402088392, 13.57577512914140748006197895243, 14.681063359256865104828160589224, 15.1387676878402390175537783121, 17.01142629120137767135063184719, 17.70350500501950014602375424790, 18.412288674800050186130164615970, 20.08685644190019537149237296115, 20.86002338408623152337079441407, 21.811420683031682999258097059914, 23.0428761563293805242606241560, 23.71630105530681387537717214527, 24.83153375359627371121746962511, 25.2749969341779633052859758091, 26.25807431022456150618503004345, 27.16104144092001504166549796586

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