↑

Properties

Label 1-231-231.32-r0-0-0
Degree $1$
Conductor $231$
Sign $0.832 - 0.553i$
Analytic cond. $1.07275$
Root an. cond. $1.07275$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 8-s + (0.5 + 0.866i)10-s − 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s − 20-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s + 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 8-s + (0.5 + 0.866i)10-s − 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s − 20-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s + 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(1.07275\)
Root analytic conductor: \(1.07275\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (0:\ ),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7997417881 - 0.2416933750i\)
\(L(\frac12)\) \(\approx\) \(0.7997417881 - 0.2416933750i\)
\(L(1)\) \(\approx\) \(0.8153508356 + 0.02550244019i\)
\(L(1)\) \(\approx\) \(0.8153508356 + 0.02550244019i\)

Euler product

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

−26.72515152878076967581491417460, −25.68985559809809159076576420619, −24.88806522763776420448430361890, −23.37837086275376303701201023997, −22.381400263203722188651652134469, −21.71400791882805926569121767218, −20.92929682375362173665208744472, −19.621959318631786727659355060534, −19.136958128383006398636784515008, −17.88502863997437663826947286526, −17.48930294081072967051204753744, −16.25156612745335621637403918316, −14.82228053528530222737425144519, −13.92857500781888176956102910803, −12.844428177475268926850715276870, −11.85555708472742866354657561582, −10.76783448458751109834935574529, −10.06310086961438213696815307219, −9.11640693288953027534647063611, −7.81514717710775491519360146790, −6.828514542354169449516323918098, −5.3234135919358794330630234869, −3.820655169118363485666801610263, −2.73648205951459483608372364499, −1.64380692377734462541434013358, 0.74345296082814047094597196714, 2.369070614109723943301941309664, 4.59298308963281948457141006530, 5.20088811635961323435502410625, 6.49758980779606466913569922948, 7.496855986525201128255938305248, 8.718459947439822909551375335974, 9.407935747200135988852752541034, 10.36821763573278094628632898023, 11.8302966931207679611075194753, 13.11880366139037290145770996747, 13.90690457247652195409153199309, 15.01129419344126054802756793488, 16.0107587153156285192826225830, 16.85372184166820280918897001270, 17.585119933664146705570278880431, 18.4913207702447252783709506193, 19.71293441918502074007370480147, 20.408116900000180238468800512560, 21.76586850227837087890905561440, 22.65661487034921512043327871036, 23.85594354730215913894268717851, 24.58311291634389902673792932983, 25.11884741359905677404197844102, 26.26059708318312151323226335486

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