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Properties

Label 1-723-723.116-r1-0-0
Degree $1$
Conductor $723$
Sign $0.00281 - 0.999i$
Analytic cond. $77.6971$
Root an. cond. $77.6971$
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  + i·2-s − 4-s + (−0.951 − 0.309i)5-s + (0.156 + 0.987i)7-s − i·8-s + (0.309 − 0.951i)10-s + (0.707 + 0.707i)11-s + (−0.891 − 0.453i)13-s + (−0.987 + 0.156i)14-s + 16-s + (−0.987 + 0.156i)17-s + (0.707 + 0.707i)19-s + (0.951 + 0.309i)20-s + (−0.707 + 0.707i)22-s + (−0.453 + 0.891i)23-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.951 − 0.309i)5-s + (0.156 + 0.987i)7-s − i·8-s + (0.309 − 0.951i)10-s + (0.707 + 0.707i)11-s + (−0.891 − 0.453i)13-s + (−0.987 + 0.156i)14-s + 16-s + (−0.987 + 0.156i)17-s + (0.707 + 0.707i)19-s + (0.951 + 0.309i)20-s + (−0.707 + 0.707i)22-s + (−0.453 + 0.891i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00281 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00281 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $0.00281 - 0.999i$
Analytic conductor: \(77.6971\)
Root analytic conductor: \(77.6971\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 723,\ (1:\ ),\ 0.00281 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2817420539 + 0.2809487856i\)
\(L(\frac12)\) \(\approx\) \(-0.2817420539 + 0.2809487856i\)
\(L(1)\) \(\approx\) \(0.5459176567 + 0.4696781742i\)
\(L(1)\) \(\approx\) \(0.5459176567 + 0.4696781742i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( 1 \)
good2 \( 1 + iT \)
5 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (0.156 + 0.987i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.891 - 0.453i)T \)
17 \( 1 + (-0.987 + 0.156i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.453 + 0.891i)T \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (0.987 - 0.156i)T \)
37 \( 1 + (-0.891 + 0.453i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (-0.891 + 0.453i)T \)
79 \( 1 + (-0.951 - 0.309i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−21.86140585748750907635857125980, −20.76679663525466422556076336306, −19.88874029956831054295740407070, −19.589984140874700374413868628688, −18.79567959571145750915292445268, −17.751864051911877629835503506516, −17.04334695661137103489142394111, −16.0978287610493189918985011451, −14.98666251236187394462506516026, −14.00691902040783517328275240282, −13.6336634062008142754791800799, −12.28818746516667525633755431747, −11.70742965624120965028112706424, −10.97869357371389470170348080958, −10.25020318276821535102004374772, −9.187203251294867544642832690, −8.33149424221991112541338898381, −7.36737718098758688766154930477, −6.453547007759718767174695120154, −4.75919676521924514945865290636, −4.26790954839956758242376001239, −3.341731586879962138431158510506, −2.346073344887040596225496141410, −0.88441213326739498138073298664, −0.123565739223409172498032037, 1.40656651486451053648901467338, 3.016692179129088560511669556671, 4.2046413366721583881830528773, 4.9084188903311744681010146146, 5.82971983940592196567808435893, 6.93152750513517294008815079541, 7.68035120235642282209703641631, 8.5165190640806730895001733739, 9.24316038106588091696456404409, 10.14977564204215498059946078927, 11.68379702558353917184891480839, 12.2150150460591562612318662834, 13.00631517080471999925195635473, 14.285228695636544165541161294533, 14.90273641622112838737858191659, 15.67579435113156875335703142648, 16.11303661341753846424921788053, 17.4228950644321380537756751677, 17.72747190064866855304811009666, 18.91377326842817815289981900440, 19.5255304005506377165260633031, 20.40638265214547485500449953712, 21.661168032532826650721551528551, 22.45169721628546410289835205567, 22.866099459132219573300984595262

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