Properties

Degree 1
Conductor 73
Sign $0.835 - 0.549i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s i·7-s + 8-s + 9-s i·10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.866 + 0.5i)14-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s i·17-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s i·7-s + 8-s + 9-s i·10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.866 + 0.5i)14-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.835 - 0.549i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.835 - 0.549i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $0.835 - 0.549i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (24, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 73,\ (0:\ ),\ 0.835 - 0.549i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3634052676 - 0.1088154495i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3634052676 - 0.1088154495i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4975545579 + 0.04754349274i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4975545579 + 0.04754349274i\)

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

−31.33860501829332766570581209489, −30.507636899704489275372694374671, −29.151646535034788535754334509609, −28.37806407008272789283606388720, −27.63303737182188125150935188633, −26.83306005827238116018222462464, −25.08759742332981502015799124775, −23.904726856236958691441773727994, −22.53178671189727644810117938046, −21.90237500185971694409361826650, −20.579248118492433735081946492600, −19.3096946371100196999653672989, −18.50679429529993773199952815650, −17.151296829020622598240301118078, −16.41247267379965200207444899086, −14.90176027592689337679870607449, −12.57328654819270183231195387237, −12.19631130077201137257770283609, −11.222576678292564016862689243382, −9.76883265106743977249876170764, −8.569943113509999250688701538528, −7.0541629301486713284230608127, −5.11110852365547772255347322085, −3.86757530674488778985387704566, −1.652654487231475692726238820130, 0.631266945954063859501442755, 4.01316532047686286463052181923, 5.37994343923226028325369470933, 7.02147709981753888683179453363, 7.42807885054769575824129548669, 9.43408077137460782326451155658, 10.72805553287902299362619030997, 11.6436723283997434740220882432, 13.47952931040345334401120005572, 14.83833770372785792476043327013, 15.97921063394072991325160071591, 16.90621862680540176633241762365, 17.79760986360941965867560329791, 19.04514913678649974828389564370, 20.00079433439226378220151546729, 22.21366601782945139821030804459, 22.85809432601192758401448738467, 23.849732458159579136883326626378, 24.6697782337390974475259061816, 26.38395028749233752314146823668, 27.18255852406004615616852687229, 27.72659948456073073857654787683, 29.268643037651605666410315174, 30.10320967234611649667614896711, 31.75591268486242532889607670117

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