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Properties

Label 1-42e2-1764.47-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.970 + 0.239i$
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  + (−0.222 + 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

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

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.970 + 0.239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.153451317 + 0.1403750531i\)
\(L(\frac12)\) \(\approx\) \(1.153451317 + 0.1403750531i\)
\(L(1)\) \(\approx\) \(0.8326596037 + 0.1581570209i\)
\(L(1)\) \(\approx\) \(0.8326596037 + 0.1581570209i\)

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

−20.06043943211645370801644010949, −19.42653834545936418973163043794, −18.53370444473296914395261633, −17.78967636170395651123948778241, −17.022871618778972186226298593146, −16.37281096173805036668177259578, −15.50253849808843967310872702756, −15.16659235381843304139346051553, −13.91024911838013218759276599096, −13.07675259754524411543253201514, −12.83922490373954890872792660874, −11.78831963803408979518210953067, −11.053005172356966496698684207564, −10.20719374743823810450391539021, −9.40868492799498610325173127115, −8.4161663903279871037949640678, −8.07464881726464010537010671132, −7.11333879682077554927949673717, −5.93077123513420770890347320551, −5.316724154187854193268452244470, −4.539304194110992701275259239203, −3.5772860353483669504892018368, −2.63368532873488420003582262965, −1.52424473937838405842787266962, −0.50673297910938581074533395105, 0.37567113168932450423466402003, 2.11273163488938825861969464363, 2.4468677238834806156822991032, 3.65296608675921348464623809949, 4.443968587161203159985042306219, 5.32385372028074003554083334591, 6.517305809357882112836112927159, 6.94737991960017737880084888761, 7.72605976756496632849408471063, 8.76849677849993125507152983147, 9.44940850807296860251961773814, 10.546390146542363977346408711393, 10.95098475330662388073329080785, 11.67133656069801739377425552937, 12.77508787004123673967660267876, 13.28642310062367591928787321926, 14.356165443667715325832960421066, 14.81768311353167500090006955308, 15.64129343287638782570936284003, 16.2279774016079708543717726251, 17.35127183225088627875245380195, 17.85814451005306447368466598289, 18.807763806749312187003994180313, 19.07941473848182003995001940031, 20.1064118652027989274608941179

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