Properties

Degree $2$
Conductor $432$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 7-s + 3·11-s − 4·13-s − 2·19-s + 6·23-s + 4·25-s + 6·29-s − 5·31-s + 3·35-s + 2·37-s − 6·41-s + 10·43-s − 6·47-s − 6·49-s + 9·53-s + 9·55-s − 12·59-s + 8·61-s − 12·65-s − 14·67-s − 7·73-s + 3·77-s − 8·79-s + 3·83-s − 18·89-s − 4·91-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s + 0.904·11-s − 1.10·13-s − 0.458·19-s + 1.25·23-s + 4/5·25-s + 1.11·29-s − 0.898·31-s + 0.507·35-s + 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 6/7·49-s + 1.23·53-s + 1.21·55-s − 1.56·59-s + 1.02·61-s − 1.48·65-s − 1.71·67-s − 0.819·73-s + 0.341·77-s − 0.900·79-s + 0.329·83-s − 1.90·89-s − 0.419·91-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{432} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.784916201\)
\(L(\frac12)\) \(\approx\) \(1.784916201\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.87832764678448, −19.27560122727968, −18.27492734373622, −17.44851848039213, −17.15717244080891, −16.32239430061735, −14.97480098127354, −14.52659916056310, −13.74872398340554, −12.90632116629600, −12.08129811431436, −11.06563679257022, −10.13919133269170, −9.405159791372799, −8.657482823158305, −7.299291757638476, −6.437605506497787, −5.432878154128450, −4.473564502288481, −2.785955677684323, −1.562861180085022, 1.562861180085022, 2.785955677684323, 4.473564502288481, 5.432878154128450, 6.437605506497787, 7.299291757638476, 8.657482823158305, 9.405159791372799, 10.13919133269170, 11.06563679257022, 12.08129811431436, 12.90632116629600, 13.74872398340554, 14.52659916056310, 14.97480098127354, 16.32239430061735, 17.15717244080891, 17.44851848039213, 18.27492734373622, 19.27560122727968, 19.87832764678448

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