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Properties

Label 1-18e2-324.67-r1-0-0
Degree $1$
Conductor $324$
Sign $-0.360 + 0.932i$
Analytic cond. $34.8186$
Root an. cond. $34.8186$
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  + (−0.0581 − 0.998i)5-s + (−0.973 − 0.230i)7-s + (0.835 + 0.549i)11-s + (−0.993 − 0.116i)13-s + (0.766 − 0.642i)17-s + (−0.766 − 0.642i)19-s + (−0.973 + 0.230i)23-s + (−0.993 + 0.116i)25-s + (0.597 + 0.802i)29-s + (0.286 − 0.957i)31-s + (−0.173 + 0.984i)35-s + (0.173 + 0.984i)37-s + (0.396 + 0.918i)41-s + (−0.893 + 0.448i)43-s + (0.286 + 0.957i)47-s + ⋯
L(s)  = 1  + (−0.0581 − 0.998i)5-s + (−0.973 − 0.230i)7-s + (0.835 + 0.549i)11-s + (−0.993 − 0.116i)13-s + (0.766 − 0.642i)17-s + (−0.766 − 0.642i)19-s + (−0.973 + 0.230i)23-s + (−0.993 + 0.116i)25-s + (0.597 + 0.802i)29-s + (0.286 − 0.957i)31-s + (−0.173 + 0.984i)35-s + (0.173 + 0.984i)37-s + (0.396 + 0.918i)41-s + (−0.893 + 0.448i)43-s + (0.286 + 0.957i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.360 + 0.932i$
Analytic conductor: \(34.8186\)
Root analytic conductor: \(34.8186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (1:\ ),\ -0.360 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2138532852 + 0.3118057197i\)
\(L(\frac12)\) \(\approx\) \(0.2138532852 + 0.3118057197i\)
\(L(1)\) \(\approx\) \(0.7839295515 - 0.1045809424i\)
\(L(1)\) \(\approx\) \(0.7839295515 - 0.1045809424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.0581 - 0.998i)T \)
7 \( 1 + (-0.973 - 0.230i)T \)
11 \( 1 + (0.835 + 0.549i)T \)
13 \( 1 + (-0.993 - 0.116i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.973 + 0.230i)T \)
29 \( 1 + (0.597 + 0.802i)T \)
31 \( 1 + (0.286 - 0.957i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (0.396 + 0.918i)T \)
43 \( 1 + (-0.893 + 0.448i)T \)
47 \( 1 + (0.286 + 0.957i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.835 - 0.549i)T \)
61 \( 1 + (-0.686 + 0.727i)T \)
67 \( 1 + (-0.597 + 0.802i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (-0.396 + 0.918i)T \)
83 \( 1 + (-0.396 + 0.918i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (-0.0581 + 0.998i)T \)
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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

−24.79431554456228618488589109445, −23.5311423508910541200466078618, −22.75966738259299280836563204965, −21.94860338288126287204694086983, −21.38724356426781927023133037930, −19.7364650914077268868203215970, −19.299429551314481612368822299450, −18.53545718592680904155043428507, −17.32336806864778521075369335382, −16.5197421169032704089122035967, −15.46219164024790060980329541948, −14.5249886900591258401345388903, −13.87675544113305518344814506060, −12.46068280669095495180346576086, −11.871184326337042362984358476528, −10.48244974657117192851568814848, −9.93370904549870585116457466533, −8.72932447112888069097639859273, −7.51497090052053915147429585602, −6.47257563233572991943068019271, −5.848061354233210300788738234019, −4.07161290111511819507027239452, −3.20810937882680803954127212820, −2.051448447536832358559986223007, −0.11653676313199111864205692607, 1.17271608048061504323695923897, 2.68763908994389791023675783444, 4.05855286152586113862546318736, 4.93285702637436836599319526805, 6.20766887773906273738846119149, 7.21684100315610476257893382786, 8.34494944252132973956335136767, 9.58502651223892772246144153334, 9.879521191594610540677814266213, 11.58930104048588295989677154590, 12.38558906680910956516836584713, 13.088608735573689789899044038624, 14.18536602864072837634970462803, 15.25781733529096359229022727047, 16.31083679185272396716934940006, 16.89877921216619885358963958029, 17.772729320212144390157524471321, 19.19012929804507438638114406671, 19.814512534219016599021480693485, 20.48454449634717078060244335632, 21.69556261494346679584392395791, 22.46298082976738320915050846222, 23.42865864198684467947277637439, 24.264601888998879127581438168987, 25.235392528616009523698162641566

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