Properties

Degree $2$
Conductor $15975$
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  − 1.94·2-s + 2.80·4-s − 3.51·8-s + 4.04·16-s − 0.445·19-s − 1.24·29-s − 4.38·32-s − 1.94·37-s + 0.867·38-s − 0.867·43-s + 49-s + 2.43·58-s + 4.49·64-s + 71-s + 0.867·73-s + 3.80·74-s − 1.24·76-s − 1.80·79-s + 1.94·83-s + 1.69·86-s + 1.24·89-s − 1.94·98-s + 0.445·101-s + 1.56·103-s − 1.24·109-s − 3.49·116-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15975 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15975\)    =    \(3^{2} \cdot 5^{2} \cdot 71\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 15975,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.4885031853\] \[L(1,\rho) \approx 0.4434992338\]

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