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Properties

Label 1-42e2-1764.1571-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.00356 - 0.999i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.623 − 0.781i)5-s + (−0.222 + 0.974i)11-s + (−0.955 + 0.294i)13-s + (0.826 + 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.0747 + 0.997i)29-s + (−0.5 + 0.866i)31-s + (0.0747 − 0.997i)37-s + (−0.988 − 0.149i)41-s + (0.988 − 0.149i)43-s + (−0.955 + 0.294i)47-s + (−0.0747 − 0.997i)53-s + (0.623 + 0.781i)55-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)5-s + (−0.222 + 0.974i)11-s + (−0.955 + 0.294i)13-s + (0.826 + 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.0747 + 0.997i)29-s + (−0.5 + 0.866i)31-s + (0.0747 − 0.997i)37-s + (−0.988 − 0.149i)41-s + (0.988 − 0.149i)43-s + (−0.955 + 0.294i)47-s + (−0.0747 − 0.997i)53-s + (0.623 + 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00356 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00356 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.00356 - 0.999i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.00356 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9609251654 - 0.9575085307i\)
\(L(\frac12)\) \(\approx\) \(0.9609251654 - 0.9575085307i\)
\(L(1)\) \(\approx\) \(1.015380559 - 0.08735899319i\)
\(L(1)\) \(\approx\) \(1.015380559 - 0.08735899319i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (-0.955 + 0.294i)T \)
17 \( 1 + (0.826 + 0.563i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.0747 + 0.997i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (-0.0747 - 0.997i)T \)
59 \( 1 + (0.988 - 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.955 - 0.294i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.25323668266827343718640623324, −19.32447040831567909130454678906, −18.80073222805274329069503502363, −18.07852838470398676921550986392, −17.28106613255100737028591819415, −16.72261023630616208379384699063, −15.72023557981497113440663976740, −14.99217828268922224843302749735, −14.28314388617239886213836462603, −13.58413014319696664760922052909, −12.990899683885231899546029513384, −11.79082233773817917032302462192, −11.32388562445227146167176510981, −10.267858716000437193660302439673, −9.865116675917739370478103228397, −8.95724787369296181841065742494, −7.89329827546932867227355399097, −7.29633884370187898727900894032, −6.27896394155286361532693751831, −5.69235961243911855717565652904, −4.818955333453008509945326222558, −3.62572793940853645165491967265, −2.79346911313662389518507944079, −2.15224783388224939494997796308, −0.805914683371406266715179873221, 0.278686640132392621938026754199, 1.7099603952262957729730692893, 2.03286521277702026376433172781, 3.40925933891668547461893322300, 4.42297422888758990365703694411, 5.10665677713404120300672768975, 5.86576771713234135532548357260, 6.81260737618999149227078426211, 7.71170260448330223789989253174, 8.46842007066895206071769377668, 9.36982979469337650711658239182, 10.03122660503182267088024047432, 10.57133734720700004743277594717, 11.91888329573704875846581580957, 12.6071245062342058689358641322, 12.781328604560919177942924268151, 14.2693269040054046341686186636, 14.36043057130008079304006387636, 15.44628700107452089743804671560, 16.47309741608354903130268530638, 16.775279662230789538792212141103, 17.721569542047800969670172191882, 18.18757782068881808606851984076, 19.288684180883550773375777449825, 19.90399921627624023801124535478

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