Properties

Degree 1
Conductor 173
Sign $0.838 + 0.544i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.976 + 0.217i)2-s + (0.934 − 0.357i)3-s + (0.905 + 0.424i)4-s + (−0.639 + 0.768i)5-s + (0.989 − 0.145i)6-s + (−0.391 + 0.920i)7-s + (0.791 + 0.611i)8-s + (0.744 − 0.667i)9-s + (−0.791 + 0.611i)10-s + (0.322 − 0.946i)11-s + (0.997 + 0.0729i)12-s + (−0.976 − 0.217i)13-s + (−0.581 + 0.813i)14-s + (−0.322 + 0.946i)15-s + (0.639 + 0.768i)16-s + (−0.957 − 0.288i)17-s + ⋯
L(s,χ)  = 1  + (0.976 + 0.217i)2-s + (0.934 − 0.357i)3-s + (0.905 + 0.424i)4-s + (−0.639 + 0.768i)5-s + (0.989 − 0.145i)6-s + (−0.391 + 0.920i)7-s + (0.791 + 0.611i)8-s + (0.744 − 0.667i)9-s + (−0.791 + 0.611i)10-s + (0.322 − 0.946i)11-s + (0.997 + 0.0729i)12-s + (−0.976 − 0.217i)13-s + (−0.581 + 0.813i)14-s + (−0.322 + 0.946i)15-s + (0.639 + 0.768i)16-s + (−0.957 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.838 + 0.544i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.838 + 0.544i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $0.838 + 0.544i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (64, \cdot )$
Sato-Tate  :  $\mu(86)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ 0.838 + 0.544i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.248427424 + 0.6661108600i$
$L(\frac12,\chi)$  $\approx$  $2.248427424 + 0.6661108600i$
$L(\chi,1)$  $\approx$  2.011814041 + 0.3834550973i
$L(1,\chi)$  $\approx$  2.011814041 + 0.3834550973i

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

−27.37976712524460358620485061327, −26.360004562595530275035115305854, −25.3178876122914514350395364742, −24.35741324494514789140487157690, −23.6452800611797783025260726380, −22.49925565278576554205575962804, −21.58425613256915421969158800013, −20.4028843757552520786129723873, −19.83407554201640490744208736228, −19.48305006231737819465922344822, −17.33195629968920610112871680210, −16.12650853114027860200469488453, −15.44107205098585768524615002467, −14.445980582870902351896007002, −13.423142483932318056701830177210, −12.67829781796541463494741902949, −11.517703521063467776147045337609, −10.16585924178741873102784927901, −9.256562415692705201432317427429, −7.64671565197952537046512904322, −6.92693795627363383334334601780, −4.82604702713187214094309510218, −4.300715711211935334279203646019, −3.17890644612534406014580087217, −1.690045271421166703416952642703, 2.35802313287894213102753824892, 3.07459973250687251684044997900, 4.178239339058304828221970425783, 5.88489076069986619152079627741, 6.913382123193342549900926452552, 7.87040175633682418463840873067, 9.004462638932913093238027542932, 10.625731272344895955001837700497, 11.95806999481980203540718396326, 12.56957019097183389781279383658, 13.95401784277108226228662404032, 14.53945519639018646742417588981, 15.48968822787543674066257456635, 16.21198554253580317856210449644, 18.01665749088405828413747782988, 19.14278601138638654107739684970, 19.72454068313844878463332331031, 20.88559857763697000049361604088, 22.10882817446619344011663346475, 22.47965043122823187028488104213, 23.9907806272551398546781673052, 24.579729438853371285714251779568, 25.426217103532941660870837372177, 26.46283116737899167270908770238, 27.17346841454239125290440017530

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