Properties

Degree $2$
Conductor $324$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s − 13-s + 16-s − 17-s − 20-s − 26-s − 29-s + 32-s − 34-s − 37-s − 40-s + 2·41-s + 49-s − 52-s + 2·53-s − 58-s − 61-s + 64-s + 65-s − 68-s − 73-s − 74-s − 80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 324,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 1.188200343\] \[L(1,\rho) \approx 1.412893501\]

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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