Properties

Degree $2$
Conductor $162$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

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

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{162} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73678\)
\(L(\frac12)\) \(\approx\) \(1.73678\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
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 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.97130443985463931376959920846, −12.30257690637103727384354789287, −10.74648774185021180091471784521, −9.880068252452829064500103261921, −9.101601246278505892226541271616, −7.24345462277249174674167705434, −6.18735538198950166736525445007, −5.49090131952205180218546644094, −3.69589098584017718744945020548, −2.30016225881827778465495235815, 2.30016225881827778465495235815, 3.69589098584017718744945020548, 5.49090131952205180218546644094, 6.18735538198950166736525445007, 7.24345462277249174674167705434, 9.101601246278505892226541271616, 9.880068252452829064500103261921, 10.74648774185021180091471784521, 12.30257690637103727384354789287, 12.97130443985463931376959920846

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