Properties

Label 1-231-231.227-r1-0-0
Degree $1$
Conductor $231$
Sign $0.540 + 0.841i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.540 + 0.841i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ 0.540 + 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220369303 + 0.6664488872i\)
\(L(\frac12)\) \(\approx\) \(1.220369303 + 0.6664488872i\)
\(L(1)\) \(\approx\) \(0.8654047463 + 0.1451849792i\)
\(L(1)\) \(\approx\) \(0.8654047463 + 0.1451849792i\)

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

−25.74298737783641802561549957516, −25.26629864806142205245496849346, −24.391450948221269188357995821826, −23.483774101743987737173810007378, −22.15888664137732897455798337535, −20.7937438084809252847775972398, −20.52129324926599781900699662217, −19.285508099729884565582139699832, −18.31682756492595333122487687218, −17.4344322073459614778253930686, −16.711456439641615438894342200159, −15.79036120824890849880892592215, −14.747969686558280456832229298238, −13.49990851810270757020402680755, −12.409006600738231563056360224607, −11.2871594625413499830119518140, −10.05694303032990732219080138764, −9.45063406671007489719016385744, −8.29487255872730981589481919033, −7.4651698470803605646887651475, −5.942517013661624288177676260553, −5.321970782499309987078914542208, −3.312119470584781038196856057931, −1.77995615047934929491801626136, −0.71464296461908714416806297176, 1.20821576293906928766296249273, 2.43953783174608007011259242798, 3.544187906223862514993798878703, 5.50886833298265933269991146865, 6.71705800596463132489001613490, 7.444194671115930726863088463103, 8.89324013636307399289049123434, 9.6465965633734475909223181855, 10.669401064180687068821939363747, 11.45853927174372279197294467877, 12.633676624391121950655021307761, 13.97818535015022851357220641673, 14.86860615611364678794250259599, 16.182422832383169466926002766707, 16.883539537829866697380100291512, 18.088919962328481288084165471272, 18.53140957384791110181895904375, 19.50726424400172980404858589693, 20.66730953056704908662457354155, 21.41722221354593852265542196492, 22.312180189666423292766814974669, 23.5857732855472247233249188915, 24.76733789040832941915584456758, 25.521044852417661201436679660430, 26.3848266164216663351250460485

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