L-function 1-165-165.68-r1-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
	11	$1$
good	2	$1 + (-0.587 + 0.809i)T$
	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	$1 + (0.951 - 0.309i)T$

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