Properties

Label 1-171-171.86-r0-0-0
Degree $1$
Conductor $171$
Sign $0.947 - 0.320i$
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 + (−0.173 + 0.984i)5-s + 7-s + (−0.5 + 0.866i)8-s + (0.939 + 0.342i)10-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)14-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + (0.5 − 0.866i)20-s + (0.939 − 0.342i)22-s + (0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.173 + 0.984i)5-s + 7-s + (−0.5 + 0.866i)8-s + (0.939 + 0.342i)10-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)14-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + (0.5 − 0.866i)20-s + (0.939 − 0.342i)22-s + (0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.160008267 - 0.1907134217i\)
\(L(\frac12)\) \(\approx\) \(1.160008267 - 0.1907134217i\)
\(L(1)\) \(\approx\) \(1.079192847 - 0.2586912868i\)
\(L(1)\) \(\approx\) \(1.079192847 - 0.2586912868i\)

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 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 - T \)
89 \( 1 + (0.173 + 0.984i)T \)
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

−27.32910641538147418793398169275, −26.840576276202738716480213709774, −25.25180805756357229325571158999, −24.69935892975039293160663717873, −23.9410322608481525948494056966, −23.10005675899116242223992555458, −21.76206309559301581850881267970, −21.06605163211100985096921951405, −19.76920914297723872649838943278, −18.60490335571643361433582339019, −17.30522647980680492759320686897, −16.89735138395270202738245757807, −15.76220805231658534527663152723, −14.714084296144917608464910923890, −13.88174811909947615618631787763, −12.67535223050978113302981242931, −11.81092550695566087380815182828, −10.1622548222842294064574949054, −8.67628713771433607287342034218, −8.26434772123588698345786366565, −6.99015682088830084375363447902, −5.43724760367491578624307181967, −4.879306690143001359313327068835, −3.4753917617465992505039615073, −1.09129891317732237234767402324, 1.63580865015412363213161047213, 2.79435082103904319016176332323, 4.15349812633067070560232896677, 5.17469266562751677821617120640, 6.86672558926380661595357360952, 8.053951197609279073112645315266, 9.53126139771410195925377303686, 10.35357109098094902993861616694, 11.60542382848006963756056840378, 11.97422303546024579335213123304, 13.58973143966402126028535357296, 14.56403247302778058597259058716, 15.045891020290792043281577992172, 17.07910727877412293892394452227, 17.8924425991169204543382900056, 18.86822229035684529026147228297, 19.63268603590165606171351811551, 20.84527925228424826401119075887, 21.54294016400126313048077044020, 22.625319234536195118896825936542, 23.26463854302644771681914857318, 24.4268088489976596047911910113, 25.75888002873548044476791167996, 26.96035696971597563252374846464, 27.40309519166235229275723093149

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