↑

Properties

Label 1-207-207.203-r0-0-0
Degree $1$
Conductor $207$
Sign $-0.186 + 0.982i$
Analytic cond. $0.961303$
Root an. cond. $0.961303$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.327 + 0.945i)2-s + (−0.786 + 0.618i)4-s + (0.580 − 0.814i)5-s + (−0.235 + 0.971i)7-s + (−0.841 − 0.540i)8-s + (0.959 + 0.281i)10-s + (0.981 − 0.189i)11-s + (0.235 + 0.971i)13-s + (−0.995 + 0.0950i)14-s + (0.235 − 0.971i)16-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (0.0475 + 0.998i)20-s + (0.5 + 0.866i)22-s + (−0.327 − 0.945i)25-s + (−0.841 + 0.540i)26-s + ⋯
L(s)  = 1  + (0.327 + 0.945i)2-s + (−0.786 + 0.618i)4-s + (0.580 − 0.814i)5-s + (−0.235 + 0.971i)7-s + (−0.841 − 0.540i)8-s + (0.959 + 0.281i)10-s + (0.981 − 0.189i)11-s + (0.235 + 0.971i)13-s + (−0.995 + 0.0950i)14-s + (0.235 − 0.971i)16-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (0.0475 + 0.998i)20-s + (0.5 + 0.866i)22-s + (−0.327 − 0.945i)25-s + (−0.841 + 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.186 + 0.982i$
Analytic conductor: \(0.961303\)
Root analytic conductor: \(0.961303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 207,\ (0:\ ),\ -0.186 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8695229598 + 1.050345712i\)
\(L(\frac12)\) \(\approx\) \(0.8695229598 + 1.050345712i\)
\(L(1)\) \(\approx\) \(1.035814014 + 0.6731318855i\)
\(L(1)\) \(\approx\) \(1.035814014 + 0.6731318855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.327 + 0.945i)T \)
5 \( 1 + (0.580 - 0.814i)T \)
7 \( 1 + (-0.235 + 0.971i)T \)
11 \( 1 + (0.981 - 0.189i)T \)
13 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.786 + 0.618i)T \)
31 \( 1 + (0.0475 - 0.998i)T \)
37 \( 1 + (-0.415 + 0.909i)T \)
41 \( 1 + (-0.580 + 0.814i)T \)
43 \( 1 + (-0.0475 - 0.998i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (-0.235 - 0.971i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (-0.723 + 0.690i)T \)
83 \( 1 + (0.580 + 0.814i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.995 + 0.0950i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.789670153748102756450757781616, −25.65902499899850564358551917884, −24.57737429303486496565201280165, −23.1885149855684061905113006919, −22.66146073899204261190973468557, −21.88511736432905198500273067013, −20.7718249542978397865085274311, −19.91985378209671434634319087558, −19.1603853666766686939027762132, −17.827742450241069430016542386663, −17.45977577422757142426827078910, −15.74426293161758632381119630170, −14.46129269542476318176968117372, −13.83546992145418691581991633205, −12.97741088861447615512752966987, −11.66585164698386139908516626784, −10.73673711401101370198484005562, −9.984237123513282045326898574908, −9.01168894767332492314046055640, −7.25728353930384864804955279292, −6.213216896232509677036670018328, −4.84538043330595891503786058544, −3.580198592493293311889505290, −2.63412293594278146591616837669, −1.06417476338786566573279067443, 1.73360429981076303824259820175, 3.61781290445421112835455205022, 4.78251698716596708298158305156, 5.9474921387816467556227136250, 6.53400959626474229951613553302, 8.28245780727469000829026983237, 8.919754837245896034343538237881, 9.81340843448683939729068277530, 11.835221978718870108321420645251, 12.52048314005947485610969096010, 13.61351922687776521981830369945, 14.477423548248595025845629842033, 15.5268057050995233988504907400, 16.58855419695306026059457112976, 17.08630253697115199163287258494, 18.30036659993908275294127633793, 19.211629475645066257590047141192, 20.69587371095915546568018759502, 21.729641048388042723637602993867, 22.146322206060224169878975487751, 23.55066980210043378673307645305, 24.32804042582535756522679628496, 25.15229249998201087906695234791, 25.67924767469066712031002659293, 26.89718605838162068636601917398

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