Properties

Degree 1
Conductor 73
Sign $-0.634 + 0.773i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 − 0.342i)2-s + (−0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + 10-s + (−0.939 + 0.342i)11-s + (0.173 − 0.984i)12-s + (0.173 + 0.984i)13-s + (0.173 + 0.984i)14-s + (0.766 + 0.642i)15-s + (0.173 + 0.984i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 − 0.342i)2-s + (−0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + 10-s + (−0.939 + 0.342i)11-s + (0.173 − 0.984i)12-s + (0.173 + 0.984i)13-s + (0.173 + 0.984i)14-s + (0.766 + 0.642i)15-s + (0.173 + 0.984i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.634 + 0.773i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.634 + 0.773i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $-0.634 + 0.773i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (16, \cdot )$
Sato-Tate  :  $\mu(9)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (0:\ ),\ -0.634 + 0.773i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.0003761642187 + 0.0007952982925i$
$L(\frac12,\chi)$  $\approx$  $0.0003761642187 + 0.0007952982925i$
$L(\chi,1)$  $\approx$  0.3154054031 - 0.1062370111i
$L(1,\chi)$  $\approx$  0.3154054031 - 0.1062370111i

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

−31.65364137396657137377235990277, −29.65306692387886602389086521987, −28.47058359356749118328307739594, −27.930078443627321353793614416944, −26.9774590981860681140879374, −26.040067216542253767669876281094, −24.78142677373471535787167462287, −23.55214146793791245975136326516, −22.557778889359616093168473701049, −21.016727408748985259820173356891, −20.06009167379724342291590174074, −18.79872742675125895608916810406, −17.7558812820232803553823482261, −16.24355710698557663543939811401, −15.82771929542678745046695249066, −14.93457232126758618414141083604, −12.52086689449267952797979858848, −11.325321784764333560376475314910, −10.29744417325347370867533898458, −8.98796722379019432113373115391, −8.0056051820264156693681495257, −6.20812263444092419308258816168, −5.02138989001083051392780279034, −3.019875924697657868091694030547, −0.00134915037009762164350461399, 2.10856525779833231432838959504, 3.95659730473704639674324233698, 6.50171810659043083810630672632, 7.358166417797208855069758069483, 8.39004411098272818686704314841, 10.31237438303233743589173329867, 11.18514900702503695802821831188, 12.30381692943762508598756647872, 13.43012849461408740185968287904, 15.45634958471339043239582741096, 16.60106090874821532995905874771, 17.56410064105683586866719926773, 18.863411049649895202749691319434, 19.36184256893714617578399268556, 20.508914017755331238317005123509, 22.15522766586045971000535812381, 23.49655581371472136808428731109, 24.09854807943550545689824276364, 25.921973503345439686488673584307, 26.356516680425618393630459480149, 27.87476453790483251237694786442, 28.622369434353501040929375822607, 29.735763687875598153516185834250, 30.48351022698545950351445950232, 31.50609331591354884854874143647

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