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Properties

Label 1-231-231.59-r0-0-0
Degree $1$
Conductor $231$
Sign $0.711 + 0.702i$
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  + (0.978 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(1.07275\)
Root analytic conductor: \(1.07275\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (0:\ ),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.158113918 + 0.8856440790i\)
\(L(\frac12)\) \(\approx\) \(2.158113918 + 0.8856440790i\)
\(L(1)\) \(\approx\) \(1.881773058 + 0.4923334666i\)
\(L(1)\) \(\approx\) \(1.881773058 + 0.4923334666i\)

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 \( 1 + (-0.309 - 0.951i)T \)
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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

−25.8233368400581555482728762844, −25.18751312767763882303184238887, −24.053750619244788169150016901079, −23.71999990047622437721428622129, −22.27344548320817385342948511723, −21.63421044095457183815987307019, −20.83236916941695974345528766623, −19.9378404460423526555601937594, −19.02487613834676248349522096876, −17.54653888187391937268171128228, −16.66036652668303864783211539363, −15.69856854154563856564950094729, −14.629240953153522205111779528151, −13.605890755350918738307400182069, −12.99341538080577906679424923865, −11.93588460803599103085163972386, −10.96768086541717203351396494150, −9.74374209008618333212446963175, −8.73404543743503775729341276070, −7.09114675514611088561898836302, −6.15509887085565666107322593011, −4.982084642167092451503613479455, −4.22238160274792152437209951208, −2.59835232756133089785280096013, −1.53737727654730508308465386634, 2.06108683341972702872920775334, 2.98651415770876639670807192220, 4.30600365454992721213217497226, 5.55786895429291753935311377051, 6.44011028972664149537921492573, 7.369720266058507179162192875561, 8.71067497220598995803040347404, 10.387724925827993586860509749412, 10.883461265503425598122068838690, 12.32869737412789787823757190483, 13.12177982445052294106586999178, 14.12531921384152430160798509381, 14.90457763777408684777878088063, 15.72546685816319477264421864059, 17.04768001338276137600629881687, 17.76187167937531412482188464823, 19.07442746733332582001682974819, 20.15720898940116980323209443652, 21.13391907448772770606416798627, 21.96580176590114351474216585416, 22.66095919965030799817074712618, 23.49713609767299000047844990030, 24.74247490788145282519129577664, 25.25380791636402511530447846441, 26.23490712930533490278471528352

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