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Properties

Label 1-165-165.68-r1-0-0
Degree $1$
Conductor $165$
Sign $0.838 - 0.544i$
Analytic cond. $17.7317$
Root an. cond. $17.7317$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.951 + 0.309i)7-s + (0.951 + 0.309i)8-s + (−0.587 + 0.809i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.587 + 0.809i)17-s + (0.309 − 0.951i)19-s − i·23-s + (−0.309 − 0.951i)26-s + (0.587 + 0.809i)28-s + (−0.309 − 0.951i)29-s + (−0.809 − 0.587i)31-s − i·32-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.951 + 0.309i)7-s + (0.951 + 0.309i)8-s + (−0.587 + 0.809i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.587 + 0.809i)17-s + (0.309 − 0.951i)19-s − i·23-s + (−0.309 − 0.951i)26-s + (0.587 + 0.809i)28-s + (−0.309 − 0.951i)29-s + (−0.809 − 0.587i)31-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.838 - 0.544i$
Analytic conductor: \(17.7317\)
Root analytic conductor: \(17.7317\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 165,\ (1:\ ),\ 0.838 - 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7127865245 - 0.2109824760i\)
\(L(\frac12)\) \(\approx\) \(0.7127865245 - 0.2109824760i\)
\(L(1)\) \(\approx\) \(0.6498590771 + 0.1468917656i\)
\(L(1)\) \(\approx\) \(0.6498590771 + 0.1468917656i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−27.43202778414766490256324558145, −26.87906196068580598903282059539, −25.64084313829831733542221281135, −25.0883324847838511478020936581, −23.353846771337031796386979111862, −22.51388815530918362428863868330, −21.64709164968030045410928531673, −20.40351107129750362425644211370, −19.79962895063423467214463531540, −18.77730537596653034945221327482, −17.87479166047557376553035493981, −16.72548258064273201452914694102, −15.99731488141083273544128505312, −14.377566876412873025381348118947, −13.11983132298753670912823680904, −12.40427892307716538815609106675, −11.234987241741232869703199457900, −10.01282952019189713108914021186, −9.4855113282027853977976398903, −7.99703219464261568062458298772, −7.06420849072086496089830128687, −5.35860340495738219440883803995, −3.70712377628485090496546928329, −2.81326292306999618145269570557, −1.08155213212734894260125585710, 0.39539307619383528883880733466, 2.29015795555943526718631819725, 4.15987249653664420198910714200, 5.608483485119631978868834143633, 6.583937787320069283993438467244, 7.60102418460610700456402345381, 8.95971988355540688201347867172, 9.6638900837309023329763510729, 10.821511857663027468099770362205, 12.29905322495025329347283579884, 13.492096763623534566197727525008, 14.623886857076006901314450995919, 15.548511631580486560090807718280, 16.54012551885048951817141345757, 17.25470822474862196224986873040, 18.61590647942497686363562123903, 19.18890422774800247340081224250, 20.199512972717190220611676637991, 21.77535622385797781785350945863, 22.62747076781672553961963800606, 23.74131841029202094433648462129, 24.50846190509447371690276027275, 25.6361334917273038669072357714, 26.22268118270117820576856725748, 27.14864905656271463849288917210

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