Properties

Degree 2
Conductor $ 2 \cdot 3^{4} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 3·5-s − 4·7-s − 8-s + 3·10-s − 13-s + 4·14-s + 16-s − 3·17-s − 4·19-s − 3·20-s + 4·25-s + 26-s − 4·28-s + 9·29-s − 4·31-s − 32-s + 3·34-s + 12·35-s − 37-s + 4·38-s + 3·40-s + 6·41-s + 8·43-s − 12·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 1.51·7-s − 0.353·8-s + 0.948·10-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 0.670·20-s + 4/5·25-s + 0.196·26-s − 0.755·28-s + 1.67·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s + 2.02·35-s − 0.164·37-s + 0.648·38-s + 0.474·40-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(162\)    =    \(2 \cdot 3^{4}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{162} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 162,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.95125291039017, −19.50714919724039, −19.09576258449694, −17.95852134851361, −16.89715321097634, −15.90300297654159, −15.72859648772053, −14.52981508316034, −12.99743328984261, −12.32705450177555, −11.28916740532210, −10.31837968175542, −9.247376955218132, −8.278948375699901, −7.162778629729287, −6.292613357088392, −4.256386653193758, −2.946514131398973, 0, 2.946514131398973, 4.256386653193758, 6.292613357088392, 7.162778629729287, 8.278948375699901, 9.247376955218132, 10.31837968175542, 11.28916740532210, 12.32705450177555, 12.99743328984261, 14.52981508316034, 15.72859648772053, 15.90300297654159, 16.89715321097634, 17.95852134851361, 19.09576258449694, 19.50714919724039, 19.95125291039017

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