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  + 1.53·2-s + 1.34·4-s + 1.78·5-s + 0.532·8-s + 9-s + 2.73·10-s − 1.37·11-s − 1.67·13-s − 0.532·16-s + 1.53·18-s + 2.40·20-s − 2.10·22-s − 0.116·23-s + 2.19·25-s − 2.56·26-s − 1.98·31-s − 1.34·32-s + 1.34·36-s − 0.573·37-s + 0.950·40-s − 1.87·41-s − 1.84·44-s + 1.78·45-s − 0.178·46-s + 49-s + 3.36·50-s − 2.25·52-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

\[L(1/2,\rho) \approx 3.123366944\] \[L(1,\rho) \approx 2.548399296\]

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