Properties

Label 1-231-231.158-r1-0-0
Degree $1$
Conductor $231$
Sign $0.997 + 0.0650i$
Analytic cond. $24.8243$
Root an. cond. $24.8243$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.997 + 0.0650i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ 0.997 + 0.0650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.553942493 + 0.1157545652i\)
\(L(\frac12)\) \(\approx\) \(3.553942493 + 0.1157545652i\)
\(L(1)\) \(\approx\) \(1.968107623 + 0.09311482707i\)
\(L(1)\) \(\approx\) \(1.968107623 + 0.09311482707i\)

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

−25.8508999327552942891141661438, −25.1116026557544732158850322284, −23.978095153851024315214228808641, −23.08781276241653614197767258726, −22.61727340554039769747146900106, −21.55821313644942679964743031097, −20.615142053461554536690445163625, −19.65292724596985003691647492199, −18.88731306571483260601128396503, −17.70780835713501617236155796948, −16.20373614430827106889854024975, −15.514886025243834868546029593606, −14.59474609719170759997687241504, −13.78878821068048779640234345230, −12.56550806272162257503468261763, −11.768348640147863335207248247328, −10.78124342632136663294235158620, −9.959571318822957067669702203, −8.061771116696954141553605422973, −7.19559958019060815963308024011, −6.03283935844736006387893397967, −4.96474588966268147251069482299, −3.52779894933167792289586921129, −2.974019076226215886667708195399, −1.166734498082141722232704805074, 1.10151633806257932999920910867, 2.787336490793667173046779658823, 4.06012403761010185628562585795, 4.85247210826716241629956544451, 6.0402354732753053137621683261, 7.2708610330928621663994425026, 8.18070697449186946461723239761, 9.45388538829479668965834982204, 11.0530674206141362492138819160, 11.87784106660090578877644816257, 12.6461346322194568522381711692, 13.72645456132417342583261046597, 14.59347910564590136614519763848, 15.80305524858418499723904135510, 16.31173868922070394131815565752, 17.28569852048269769407142237452, 18.86480519732010358550101979294, 19.78609581406016044955267057019, 20.79500821204815209756806958708, 21.33384856755694162733212027086, 22.69085129220305916151174199400, 23.27397987099181738303548337394, 24.26830502184751956442514919663, 24.78800290000159055777786379841, 25.99555830483371699216104117447

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