Properties

Label 1-171-171.122-r0-0-0
Degree $1$
Conductor $171$
Sign $0.968 - 0.248i$
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  + 2-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 8-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 + 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s − 23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯
L(s)  = 1  + 2-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 8-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 + 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s − 23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.103554469 - 0.2655763069i\)
\(L(\frac12)\) \(\approx\) \(2.103554469 - 0.2655763069i\)
\(L(1)\) \(\approx\) \(1.863451282 - 0.1449504973i\)
\(L(1)\) \(\approx\) \(1.863451282 - 0.1449504973i\)

Euler product

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

−27.569000035167581470631659456157, −26.26164524164867010750202466801, −25.63568547659035447759537466785, −24.6254227807963775963822085819, −23.516164478504226695579148206135, −22.44615363832268513213056881610, −22.27375565579445873858891916314, −20.8358505093526936156368874820, −20.02159903374937188485584536430, −19.02579464591676608655230131312, −17.56283126396168340794952677374, −16.68499954289630494073481694882, −15.452861443140562849756708328173, −14.39086472902957831057965626387, −13.86788306063214192041746784171, −12.65038314217902083872412522163, −11.66318828412466313636412916753, −10.36820013697535896082289525981, −9.72834577947213229398690611706, −7.36491501757496536646193756007, −6.95183041467714713105258757463, −5.66501987204616368742550039891, −4.33166804187921626257990212748, −3.18230202885807135728012061003, −1.968215207828468557155961739337, 1.72399943144409944274218957458, 3.03730046624526561527392035086, 4.425772753337640823312593509789, 5.65517161237665517209556014684, 6.24916765907607066645798619868, 7.9351486425738384405306395483, 9.179813285145430970094426208949, 10.34867719488542097307859649398, 11.94810463462109686579825296844, 12.403540956316497936361463367547, 13.5092954408724282328946997410, 14.4404484794891417833672700443, 15.59173888488771617658182126020, 16.49497679289476517359029802854, 17.34922913764360952362766546277, 19.081088743634629287467629320182, 19.80052965213090754648477635255, 21.04813150778880588729986145300, 21.75397644611377115095245955923, 22.41998835356249114468396875093, 23.82754735702560684740981480465, 24.552672463456854273498015449274, 25.18348795476053126937652287281, 26.24690426484850076569653645955, 27.82783567751323419602646012737

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