↑

Properties

Label 1-147-147.62-r0-0-0
Degree $1$
Conductor $147$
Sign $0.886 - 0.462i$
Analytic cond. $0.682665$
Root an. cond. $0.682665$
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  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + (0.222 − 0.974i)8-s + (0.222 + 0.974i)10-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 19-s + (0.623 + 0.781i)20-s + (0.623 − 0.781i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.623 − 0.781i)26-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + (0.222 − 0.974i)8-s + (0.222 + 0.974i)10-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 19-s + (0.623 + 0.781i)20-s + (0.623 − 0.781i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.623 − 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ 0.886 - 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.772443044 - 0.4345505548i\)
\(L(\frac12)\) \(\approx\) \(1.772443044 - 0.4345505548i\)
\(L(1)\) \(\approx\) \(1.642380677 - 0.3096960682i\)
\(L(1)\) \(\approx\) \(1.642380677 - 0.3096960682i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 - T \)
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

−28.23382595226720003964807060199, −27.37309350357051142178861534203, −25.88991582714351610506847542911, −25.1739899038728539659833151040, −24.22218631129441987243549559320, −23.41644880487070433700597476197, −22.51458291527984460104840026429, −21.31632983193704780744740876483, −20.546868651926471480221145300206, −19.64996325571030751507472957521, −18.05000659667752894439958875476, −16.657440311006426032111806227, −16.33467373101592103613286978441, −14.99487871866829806499958524279, −14.05273575022635708300334657786, −12.90043193064538043903077220860, −12.13962386051325338776155646445, −11.097886801537531757216029364434, −9.25420045444633604261414454870, −8.24733754589803007801150111357, −6.95992938503860655868101111500, −5.79043481381725513165128201904, −4.56116014635453763937885077734, −3.68540621357522467930698629452, −1.784375967298688477878931395309, 1.71043143061467810415429965102, 3.290242405745115157092872278020, 4.01377915069000616390472660979, 5.838092427764020517253532032460, 6.53610519664868786359024008674, 8.005778629481357377911186659119, 9.74832654322376483101260603, 10.87596990590084727652613097386, 11.54063712035055166657130778596, 12.80936707335502328418075876259, 13.90731568829166836511547367443, 14.78078556704341351600103515830, 15.586758102222255719451435555194, 16.943426250414852844220221705684, 18.47719395321929289606853121757, 19.247134951723463090958284345824, 20.194428238456768290122306873728, 21.4847030283637028266679016180, 22.093010013287606779620159339144, 23.15711832173001507602478853887, 23.78037364852270454225634367337, 25.15121149580987668791432551245, 25.930669565671033854271062518418, 27.4059189827604527703632855411, 28.08769773987223482091903194118

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