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Properties

Label 1-432-432.59-r0-0-0
Degree $1$
Conductor $432$
Sign $0.858 + 0.512i$
Analytic cond. $2.00619$
Root an. cond. $2.00619$
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  + (−0.642 − 0.766i)5-s + (−0.939 + 0.342i)7-s + (−0.642 + 0.766i)11-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.984 + 0.173i)29-s + (0.939 + 0.342i)31-s + (0.866 + 0.5i)35-s + (0.866 − 0.5i)37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.939 + 0.342i)47-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)5-s + (−0.939 + 0.342i)7-s + (−0.642 + 0.766i)11-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.984 + 0.173i)29-s + (0.939 + 0.342i)31-s + (0.866 + 0.5i)35-s + (0.866 − 0.5i)37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.939 + 0.342i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.858 + 0.512i$
Analytic conductor: \(2.00619\)
Root analytic conductor: \(2.00619\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (0:\ ),\ 0.858 + 0.512i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9304159072 + 0.2565704429i\)
\(L(\frac12)\) \(\approx\) \(0.9304159072 + 0.2565704429i\)
\(L(1)\) \(\approx\) \(0.8801811195 + 0.03838123612i\)
\(L(1)\) \(\approx\) \(0.8801811195 + 0.03838123612i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.642 + 0.766i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.984 + 0.173i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.984 - 0.173i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−23.75541676704053932316573274115, −23.104965425167551049962903206584, −22.6430391815536395328907179361, −21.428034583457728472622810739725, −20.69307508371399181270718704794, −19.50950737781365379528521876197, −18.88305544226182766476832790699, −18.330260138535542353694220962161, −16.942885275156099658802353809480, −16.06484783785369974963353491454, −15.54443286465825329404971439179, −14.340938977342854015525871121969, −13.54597048135106486094593122386, −12.63176740398958715828858129155, −11.48680335589420628618859489490, −10.72593389854873447799240627074, −9.92401256413523094460794571578, −8.64073547876040577885035552783, −7.75938253325781857239017031067, −6.67094053512058143986640410532, −6.012145633759198417290812426120, −4.46992666058820523740941026303, −3.37740484155772654387841191946, −2.72354969249249427031547027236, −0.690787087842972424733604266779, 1.08064247735032776889735235513, 2.66487695703207021817382383522, 3.78154534211902205580209644165, 4.78464832887624389075256052487, 5.89291294098634879987066180975, 6.920235856496933119061049988356, 8.119221289722486019868549547267, 8.80704500749883054408090399950, 9.86875672056518040188221113135, 10.83876830013657710087163614457, 11.97699595513013060195395768105, 12.8893475109989988060705411994, 13.199247414438868160152559079851, 14.84500709143208587735550650964, 15.595144612952834332984862401435, 16.210095492126336350924826121212, 17.184029739223847012098065091183, 18.17412569988579697831705209876, 19.3155777150771919373221813762, 19.662109068012805183937151545125, 20.894251068043362915269796350700, 21.38239005951492259879339170383, 22.811960182800382598175713912092, 23.26971541402760806687305554472, 24.01372769077950541016145468900

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