Properties

Degree $2$
Conductor $9664$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 1.80·5-s + 9-s + 0.445·11-s − 0.445·17-s − 1.24·19-s + 2.24·25-s − 1.24·29-s − 1.80·31-s + 0.445·37-s + 1.80·43-s + 1.80·45-s + 1.24·47-s + 49-s + 0.801·55-s + 1.80·59-s + 81-s − 0.801·85-s − 2.24·95-s + 1.24·97-s + 0.445·99-s − 0.445·103-s − 0.801·121-s + 2.24·125-s − 1.80·127-s − 1.80·137-s − 1.24·139-s − 2.24·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9664 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9664\)    =    \(2^{6} \cdot 151\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 9664,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 2.326577924\] \[L(1,\rho) \approx 1.541404353\]

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