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Properties

Label 1-42e2-1764.43-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  + (−0.988 + 0.149i)5-s + (0.733 − 0.680i)11-s + (0.955 + 0.294i)13-s + (−0.900 − 0.433i)17-s − 19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (0.0747 + 0.997i)29-s + (0.5 − 0.866i)31-s + (−0.900 − 0.433i)37-s + (−0.988 + 0.149i)41-s + (0.988 + 0.149i)43-s + (0.733 − 0.680i)47-s + (−0.900 + 0.433i)53-s + (−0.623 + 0.781i)55-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)5-s + (0.733 − 0.680i)11-s + (0.955 + 0.294i)13-s + (−0.900 − 0.433i)17-s − 19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (0.0747 + 0.997i)29-s + (0.5 − 0.866i)31-s + (−0.900 − 0.433i)37-s + (−0.988 + 0.149i)41-s + (0.988 + 0.149i)43-s + (0.733 − 0.680i)47-s + (−0.900 + 0.433i)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: $\chi_{1764} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.00356 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7315144942 + 0.7289135448i\)
\(L(\frac12)\) \(\approx\) \(0.7315144942 + 0.7289135448i\)
\(L(1)\) \(\approx\) \(0.8634850523 + 0.03843020987i\)
\(L(1)\) \(\approx\) \(0.8634850523 + 0.03843020987i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (0.733 - 0.680i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.0747 + 0.997i)T \)
29 \( 1 + (0.0747 + 0.997i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (-0.988 + 0.149i)T \)
43 \( 1 + (0.988 + 0.149i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (-0.365 - 0.930i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.955 + 0.294i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
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

−19.892886287253432897399696722709, −19.160524118141337485343286228180, −18.58585363800367428696452698813, −17.47312857791549258402535956471, −17.09059965009413025310472666235, −15.95330220484924396360974489295, −15.53021133476386510947972582748, −14.82165259375638338583717859980, −13.989357649903146296307711561758, −12.994030391885523664746301141239, −12.3955865078972978487701762914, −11.669758221082735624870453925052, −10.84786360670681000586344072149, −10.25504515541157979638933342650, −8.95189620046164449714913067038, −8.56334059693369344618662356187, −7.732325610957739765184195251056, −6.689654532401684253039670541, −6.26287235985054166179248603130, −4.8581339990644327865074784863, −4.22413724853025525269791642449, −3.57208040473597377369046105968, −2.39530366711468715954262817752, −1.34051135664320535599684800427, −0.253106408042297087339524617464, 0.77449606518234394124170978195, 1.85512896163180813981040810574, 3.097303147895408653855824244701, 3.840723670553868586628583144390, 4.44902672361684864491571508250, 5.62014864294228075351714966049, 6.56011989958804119846988434643, 7.10538173867625878450173610834, 8.23660329643508942389494029089, 8.70417423630491303303311992021, 9.50617766622828987894657666673, 10.85897450689948992146642117167, 11.110410316688528571298106221203, 11.90601712395928190006924839916, 12.711173578198889146601968138343, 13.67369499591502717386841926109, 14.21220023309768107716402894676, 15.325122131532423256557821211297, 15.646444845977423270725586306229, 16.52800501879819103509224065349, 17.191451265403734539627339114431, 18.17637198854065719787072463513, 18.88859825484892305153417691530, 19.458752077721234335794858302732, 20.12281883061416388117140168956

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