Properties

Degree 1
Conductor 73
Sign $0.927 + 0.373i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.927 + 0.373i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.927 + 0.373i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $0.927 + 0.373i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (64, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (0:\ ),\ 0.927 + 0.373i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9410416867 + 0.1824016528i$
$L(\frac12,\chi)$  $\approx$  $0.9410416867 + 0.1824016528i$
$L(\chi,1)$  $\approx$  1.007438692 + 0.1993754872i
$L(1,\chi)$  $\approx$  1.007438692 + 0.1993754872i

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

−31.15553797638683711770299687269, −30.34051292645008162247575215710, −29.789631235959694182854531172043, −27.85257746555381096560850328700, −27.32813582931822567816888149137, −26.17853168183116024312716775420, −25.46846712415782882622518601743, −23.82936169736425687330353899379, −22.432841731337003076643078146322, −21.239305605646378376944332655559, −20.35500109715120011705491412705, −19.3899880735064161850010639041, −18.3881436022799132407780897818, −17.47684029479902530421887497852, −15.43453553254422151351743447951, −14.560117871655700864455908899779, −13.21258377790860175032007162889, −11.89419632608690371449419062097, −10.60839548784293371459828935510, −9.625831656541843497849012177422, −7.92906945612528956194629492225, −7.57097805563327699201604114370, −4.577370201522971848124573512, −3.17766410448070010016995199459, −2.01668485661567767382034843840, 1.60642668670275290712781566834, 4.08553503783596115196254133370, 5.35327045338311684881736376023, 7.3883683645836790468019409348, 8.28294768707544917572032769288, 9.011646037015392759168497725, 10.55088199714902184236459744746, 12.4135206552701795225515779001, 13.990530248514427279043360432886, 14.65541257912965393705691196270, 16.00632723097284044591235070469, 16.80523466507783422931016324695, 18.43347361306306250176439361733, 19.240476931000344304768758917366, 20.46202328816979213717894676632, 21.4561363789269550684514257228, 23.55935337006795795017268314164, 24.21285121942538884597019088393, 24.9808064378093677868592759805, 26.262455300384007488585696822612, 27.152368275994982443104124864920, 27.875809308446503237819739950901, 29.35981542609931439095372345416, 31.03663066977544343650550104839, 31.75763361427223077483613646744

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