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Properties

Label 1-42e2-1764.311-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.723 - 0.690i$
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  + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)11-s + (−0.365 + 0.930i)13-s + (−0.733 + 0.680i)17-s + (−0.5 + 0.866i)19-s + (−0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (−0.955 − 0.294i)29-s + (−0.5 + 0.866i)31-s + (0.955 + 0.294i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (−0.365 + 0.930i)47-s + (−0.955 + 0.294i)53-s + (−0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)11-s + (−0.365 + 0.930i)13-s + (−0.733 + 0.680i)17-s + (−0.5 + 0.866i)19-s + (−0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (−0.955 − 0.294i)29-s + (−0.5 + 0.866i)31-s + (0.955 + 0.294i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (−0.365 + 0.930i)47-s + (−0.955 + 0.294i)53-s + (−0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.723 - 0.690i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.723 - 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3210108102 + 0.8010943399i\)
\(L(\frac12)\) \(\approx\) \(-0.3210108102 + 0.8010943399i\)
\(L(1)\) \(\approx\) \(0.7187946564 + 0.3425479994i\)
\(L(1)\) \(\approx\) \(0.7187946564 + 0.3425479994i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (-0.365 + 0.930i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.955 - 0.294i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
41 \( 1 + (0.826 + 0.563i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (-0.955 + 0.294i)T \)
59 \( 1 + (-0.826 + 0.563i)T \)
61 \( 1 + (0.733 - 0.680i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.988 + 0.149i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.365 - 0.930i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
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

−19.769984976120853826457249766064, −18.926930137396926243057643293661, −18.21809612591840117652630944053, −17.274868591806600699187850160549, −16.59185706048700026973251955336, −15.96504504302937506943991310560, −15.12711616785238748680236044750, −14.61497672867144150140023216375, −13.47824323626198484654045580378, −12.88636792470934930096656790824, −12.109255378635421073757728865940, −11.22397063428187560540668308431, −10.878469829891519970104430174519, −9.59723160758819206000742335948, −8.87972676842783640896357195831, −8.21914749669063400281138509593, −7.39127854101853519734827213130, −6.60010766416329170001398019384, −5.58652275057870594086767591632, −4.724077606567137313381196513745, −3.96346439877661868674329201561, −3.10416958983852077938351862385, −2.10501760069938881278293036692, −0.62248839623604668983951716292, −0.243347917534642590599500518647, 1.40544993351478443048089160227, 2.19061335330788180742381950690, 3.4286180264487523274028992901, 4.12683649565086752561978119482, 4.73975422480489678593329189932, 6.10469198106758988218658923451, 6.73365152638369184824137538815, 7.54814376766495825919182542847, 8.211125237734713551981772564143, 9.25957457860259415100514775530, 9.85172318474678911276443168547, 10.9948910261493661473005259557, 11.417029452710907126606677631341, 12.30714371317768837469076608406, 12.85770102333211583372133289831, 14.04112614408301602256392311719, 14.70735855634567766252400829165, 15.19006926735414206331276535643, 16.084288697906026709432501304917, 16.81014654857441052874577058486, 17.55462183817532723684345870705, 18.40825783223375733403223332318, 19.14629983578625486925118877675, 19.74430014240529477485574487447, 20.26846312466681622873833990005

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