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Properties

Label 1-42e2-1764.1543-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.239 + 0.970i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (0.222 + 0.974i)5-s + (0.900 + 0.433i)11-s + (−0.0747 − 0.997i)13-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.365 + 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.365 + 0.930i)37-s + (0.733 − 0.680i)41-s + (0.733 + 0.680i)43-s + (0.0747 + 0.997i)47-s + (0.365 − 0.930i)53-s + (−0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)5-s + (0.900 + 0.433i)11-s + (−0.0747 − 0.997i)13-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.365 + 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.365 + 0.930i)37-s + (0.733 − 0.680i)41-s + (0.733 + 0.680i)43-s + (0.0747 + 0.997i)47-s + (0.365 − 0.930i)53-s + (−0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.239 + 0.970i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.239 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.330216445 + 1.041569980i\)
\(L(\frac12)\) \(\approx\) \(1.330216445 + 1.041569980i\)
\(L(1)\) \(\approx\) \(1.131182314 + 0.3068243076i\)
\(L(1)\) \(\approx\) \(1.131182314 + 0.3068243076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.0747 - 0.997i)T \)
17 \( 1 + (0.988 + 0.149i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (0.733 + 0.680i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.0747 - 0.997i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
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

−20.017045252148917555147427151347, −19.41321794541565017459727579009, −18.77499781878096708157948016362, −17.6718974458039751064802042780, −17.0757684685208393024314401385, −16.47810685702578184683598061784, −15.84893429064799380727077634506, −14.82563835971423895667193730400, −14.02883815717973745744788945643, −13.46686984524279808816018548048, −12.59016158922842842177455386411, −11.7860220644622327634446458948, −11.35162753860340015901430207060, −10.09327387451043119161168811323, −9.3271251465154057310006206956, −8.89653974768350549293458155902, −7.938895685576278580156561137740, −7.087318653347716870044396295862, −6.03922762538427753340123211643, −5.51299817131141404228336154466, −4.29294755853882666621865926496, −3.960215728245390556891551907418, −2.54248654528711146600225246330, −1.61503051864660614592827839757, −0.66870718782332629593361430727, 1.169025778632476586564575646912, 2.16366845563880527588806789390, 3.16522478585791385427452936624, 3.79221482754466572073905037889, 4.90867459539641122346975773794, 5.95691428775120674902823877199, 6.46280646698338068864153004298, 7.431863614551538514391995468447, 8.07725836508726841597205179898, 9.106604458612537831678657298524, 10.0888066270195076672629538387, 10.425248202411404754567298441327, 11.32642605309320528007637254116, 12.333589366566959370100768582933, 12.71314629494106289763180688998, 14.03053238902470433763389462218, 14.47468798117840048262954805798, 14.97026222589021418489361878896, 15.98476348955152368586200189719, 16.75204155090265246079425765548, 17.61999073369041392881398737960, 18.106081630426088157707732657305, 18.9457342170035023036417090957, 19.5687785183488895118501121120, 20.393866806582354455265891222617

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