Properties

Degree $2$
Conductor $1999$
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  − 1.87·2-s + 2.53·4-s − 1.37·5-s − 2.87·8-s + 9-s + 2.57·10-s + 1.19·11-s + 1.94·13-s + 2.87·16-s − 1.87·18-s − 3.47·20-s − 2.24·22-s − 0.573·23-s + 0.883·25-s − 3.65·26-s − 1.67·31-s − 2.53·32-s + 2.53·36-s − 1.98·37-s + 3.95·40-s + 0.347·41-s + 3.02·44-s − 1.37·45-s + 1.07·46-s + 49-s − 1.66·50-s + 4.92·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1999 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1999\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 1999,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.4765581646\] \[L(1,\rho) \approx 0.4796640746\]

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