↑

Properties

Label 1-171-171.14-r0-0-0
Degree $1$
Conductor $171$
Sign $-0.226 - 0.973i$
Analytic cond. $0.794120$
Root an. cond. $0.794120$
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  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 + 0.642i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.766 − 0.642i)10-s − 11-s + (−0.766 − 0.642i)13-s + (0.766 − 0.642i)14-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + (0.5 − 0.866i)20-s + (−0.173 − 0.984i)22-s + (0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 + 0.642i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.766 − 0.642i)10-s − 11-s + (−0.766 − 0.642i)13-s + (0.766 − 0.642i)14-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + (0.5 − 0.866i)20-s + (−0.173 − 0.984i)22-s + (0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.226 - 0.973i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ -0.226 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01455150485 + 0.01832694323i\)
\(L(\frac12)\) \(\approx\) \(0.01455150485 + 0.01832694323i\)
\(L(1)\) \(\approx\) \(0.5110222843 + 0.2656830117i\)
\(L(1)\) \(\approx\) \(0.5110222843 + 0.2656830117i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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

−28.02493185052929374759599068068, −27.09202123732363173534321426982, −26.18509511533228785346471555080, −24.68235691207657816500254357136, −23.81264418828852934856524860186, −22.84600566755666685664525467008, −21.91773332440046401021563368997, −20.95689639562489173560992869036, −20.069260484024503600835642333398, −19.10362277851848633862774025559, −18.48123528451503606061038748623, −17.10410659768018314827588344278, −15.82440573443935772337368297338, −14.97851251004152654002668759216, −13.47834079960953107961301854862, −12.63207888010854471069133601326, −11.8557068493515204239088568206, −10.87475020179308325381429396789, −9.4181799005848297581331159468, −8.80955304032580355455157711522, −7.38319461634831683148877664026, −5.49108280160500008965229533665, −4.64336724758746083563458852264, −3.26239840953806577422403015056, −2.078510168165918908656994920799, 0.01714938713934720387468219815, 3.02301842551015574964946845090, 4.1003131735355063957578918889, 5.34865453579545785341735487973, 6.83785615973596732647962800084, 7.42136929746255472847332929186, 8.48740987329539334033155844061, 10.00859092483705752492459599613, 10.930039825067131697696837803882, 12.620385346852594375404906755939, 13.32170984674612507967025198400, 14.64967293524355607273705741442, 15.32508689103626847276544413913, 16.29496793345671805512629021129, 17.27147027191491839917399821335, 18.33198243027128368976614583304, 19.29495330140177333629890459796, 20.36458753425107111930173893674, 21.896113542944813699775813191689, 22.685260816839054674451337845903, 23.470697282550491725334243727177, 24.17460041298007554124454931516, 25.45559093668771287180648141108, 26.48039460707851021125847291057, 26.74878672653763613841863293229

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