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Properties

Label 1-73-73.16-r0-0-0
Degree $1$
Conductor $73$
Sign $-0.634 + 0.773i$
Analytic cond. $0.339010$
Root an. cond. $0.339010$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.634 + 0.773i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ -0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0003761642187 + 0.0007952982925i\)
\(L(\frac12)\) \(\approx\) \(0.0003761642187 + 0.0007952982925i\)
\(L(1)\) \(\approx\) \(0.3154054031 - 0.1062370111i\)
\(L(1)\) \(\approx\) \(0.3154054031 - 0.1062370111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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