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Properties

Label 1-231-231.38-r0-0-0
Degree $1$
Conductor $231$
Sign $0.520 + 0.853i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.913 − 0.406i)5-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (0.809 − 0.587i)13-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (−0.809 + 0.587i)20-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (0.669 + 0.743i)26-s + (−0.309 + 0.951i)29-s + (−0.913 − 0.406i)31-s + (0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.913 − 0.406i)5-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (0.809 − 0.587i)13-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (−0.809 + 0.587i)20-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (0.669 + 0.743i)26-s + (−0.309 + 0.951i)29-s + (−0.913 − 0.406i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.520 + 0.853i$
Analytic conductor: \(1.07275\)
Root analytic conductor: \(1.07275\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (0:\ ),\ 0.520 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.187633977 + 0.6670201883i\)
\(L(\frac12)\) \(\approx\) \(1.187633977 + 0.6670201883i\)
\(L(1)\) \(\approx\) \(1.100498637 + 0.4773642090i\)
\(L(1)\) \(\approx\) \(1.100498637 + 0.4773642090i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−26.33347176255054658945204918101, −25.416047550893941866364552681475, −24.24142776519482844413346131811, −23.0500455808181030832708762961, −22.36819407899150131019090177621, −21.36691304627638644960481724926, −20.808223828568311802363957710646, −19.72842672927087930252576106530, −18.59903361973511897153527149626, −18.05847828677293346510445512016, −17.06681585669382080671121579384, −15.705536656142054976996158998, −14.25433071047199095872877264205, −13.75394354320559346662974711167, −12.815587248379961171804537180440, −11.49264521849421173545755186764, −10.87072992187709166598120285402, −9.54806630035165360436576612285, −9.1492317945298682306409367669, −7.46834803099728546776262625254, −6.02624646500514886804488266523, −5.03880398134557642369302210858, −3.62120287292512545697270023472, −2.52343911390746956453510189581, −1.33472290678443877391488343171, 1.310239412304828840887136387820, 3.25213522657232829770262973740, 4.660550433319765448629552858057, 5.70491619907814681676843837602, 6.43422600396930353908265205356, 7.81387574678541699950255981221, 8.77405659062495139292705404911, 9.67153885168873537982784856291, 10.81498151244824936645492410640, 12.59974707755305872895108696691, 13.17739644348883041581911495383, 14.190056780297127056972795805175, 15.071137983603217438972315754576, 16.24564287427762093992533813460, 16.901250817204312403401536316595, 17.95275032739260857073511933761, 18.513485284945463954610960584413, 20.06204152385977491193571634196, 21.097540327679162125779585996197, 22.022006709181931201751466766, 22.84005893262743053272139042603, 23.96536920223760158323954519729, 24.64794744473016634669717612813, 25.57262441588374024410643388531, 26.120390293760893560650450063363

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