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Properties

Label 1-6e3-216.131-r0-0-0
Degree $1$
Conductor $216$
Sign $0.286 + 0.957i$
Analytic cond. $1.00309$
Root an. cond. $1.00309$
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  + (−0.939 − 0.342i)5-s + (−0.173 + 0.984i)7-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (0.766 + 0.642i)29-s + (−0.173 − 0.984i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (−0.766 + 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.173 − 0.984i)47-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (−0.173 + 0.984i)7-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (0.766 + 0.642i)29-s + (−0.173 − 0.984i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (−0.766 + 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.173 − 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.286 + 0.957i$
Analytic conductor: \(1.00309\)
Root analytic conductor: \(1.00309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (0:\ ),\ 0.286 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6925744515 + 0.5156025756i\)
\(L(\frac12)\) \(\approx\) \(0.6925744515 + 0.5156025756i\)
\(L(1)\) \(\approx\) \(0.8517480670 + 0.1945158302i\)
\(L(1)\) \(\approx\) \(0.8517480670 + 0.1945158302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−26.7279315033830936152371341831, −25.46620064156618263348194304237, −24.522235532987504649309295239443, −23.39563387696644948768382675493, −22.82947037881612012149602275129, −21.94131091914833647428301785297, −20.45370903224849885772304412689, −19.8240245933309687128692864943, −19.0825393816681884395554634603, −17.76993312263015298563947058307, −16.881048890457260528281148794302, −15.905571179018273711221043764278, −14.8236295965476534505879335321, −14.06883486551345346394789232792, −12.71711769236244684965961783826, −11.82183392049641359614283299251, −10.768755349428564278812071690850, −9.83440655598775446090394048400, −8.47743975153733313650908306071, −7.28275030448215443277089640764, −6.7323670580760245617065540652, −4.87054186535453912521649754095, −3.932215698454242952376193994678, −2.74764347000779865601477121470, −0.68330161950506927654432823185, 1.61230794520791665876724312240, 3.260701821426860401683829919680, 4.322338499067723544355049716995, 5.62741114065654846108392220734, 6.79482487097882247300153806409, 8.11150987998421936254146779036, 8.89512344193613772090104667844, 10.01783276290446228657514321315, 11.64731789268263039469069509130, 11.96645948058190819916621065208, 13.08493243153843330248376527572, 14.62557692637393520725571974544, 15.14843277458497863234992381263, 16.417381128015506320161924724562, 17.009015432161193071693747303126, 18.55177287320310017477319845101, 19.271937636657588824806075524738, 19.948051655423229201567862726146, 21.36479862527810242981891538132, 21.98513813413858131199806484498, 23.14497766335794920792276483792, 24.01081612577811036644128335135, 24.86685127979513743173749695410, 25.747566727306049973080144374675, 27.06627686918629877971763837184

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