↑

Properties

Label 1-231-231.95-r0-0-0
Degree $1$
Conductor $231$
Sign $-0.959 - 0.282i$
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  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.669 − 0.743i)5-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (−0.309 − 0.951i)13-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.913 + 0.406i)19-s + (−0.309 − 0.951i)20-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (0.104 + 0.994i)26-s + (−0.809 + 0.587i)29-s + (0.669 − 0.743i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.669 − 0.743i)5-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (−0.309 − 0.951i)13-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.913 + 0.406i)19-s + (−0.309 − 0.951i)20-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (0.104 + 0.994i)26-s + (−0.809 + 0.587i)29-s + (0.669 − 0.743i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.959 - 0.282i$
Analytic conductor: \(1.07275\)
Root analytic conductor: \(1.07275\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (0:\ ),\ -0.959 - 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03972396870 - 0.2759229660i\)
\(L(\frac12)\) \(\approx\) \(0.03972396870 - 0.2759229660i\)
\(L(1)\) \(\approx\) \(0.4566863525 - 0.1703823945i\)
\(L(1)\) \(\approx\) \(0.4566863525 - 0.1703823945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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

−26.695374809452278200745366770296, −26.11592462862402801283625902743, −25.09236686224324587338562293318, −24.01534377047986785424122138323, −23.34303875481722736111682835533, −22.07412080595375130283072306764, −21.05804693134401664498470320406, −19.78836810876782071956557042206, −19.25256336354882300780386405079, −18.38856715520921698994440647193, −17.416155087466779588559903819437, −16.46895139212599911170387398402, −15.38065621713558513756073125701, −14.853887947545579808856595405069, −13.502449321405263937670383088138, −11.8384348363689370366777448582, −11.27196042310138966091252911493, −10.242351511132258849836955088561, −9.13767555243594670472736978564, −8.14115898573456870056715164159, −7.01592389344939537478850047499, −6.429790161356429909444373943957, −4.67405832649365890841021959869, −3.12538638297403467629132287381, −1.87246529727064314352597778607, 0.261893005204820921114699949917, 1.87518109507925747499299188865, 3.32969397544586627141152800273, 4.67080022329370631673241581166, 6.20453846057409803787900162171, 7.4685826380735679143240238849, 8.35479080524039412914817066589, 9.119261573998057717415947615797, 10.39874733989057704357785575101, 11.240765025002734655040587912027, 12.39717525561057992630139522272, 13.01664960578344067011442537251, 14.92779371748177249635627774491, 15.6208701526171362653652848589, 16.69856590615208639447388311638, 17.33646600344858350597621253185, 18.491931437244340151015573870168, 19.41054222232014107997873874641, 20.20290541806173755898079992099, 20.85147344254330947650206670850, 22.07536237802119393184393022852, 23.266780134606799120860346253891, 24.43049222927643304655958269333, 24.918904033471143054487404635, 26.1153422698576280858286811975

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