Properties

Degree $2$
Conductor $28096$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

Learn more

Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 1.61·5-s − 1.61·7-s + 9-s − 0.618·11-s − 2·13-s − 0.618·19-s + 1.61·25-s − 0.618·29-s − 2.61·35-s + 1.61·45-s + 1.61·49-s − 0.618·53-s − 0.999·55-s + 1.61·61-s − 1.61·63-s − 3.23·65-s + 2·71-s + 0.618·73-s + 0.999·77-s + 81-s + 1.61·83-s + 3.23·91-s − 0.999·95-s − 0.618·99-s + 2·103-s + 1.61·109-s + 0.618·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28096 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28096\)    =    \(2^{6} \cdot 439\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 28096,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 1.525069628\] \[L(1,\rho) \approx 1.080427575\]

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