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  + 1.56·2-s + 1.44·4-s + 0.695·8-s − 0.356·16-s + 1.24·19-s + 1.80·29-s − 1.25·32-s + 1.56·37-s + 1.94·38-s − 1.94·43-s + 49-s + 2.81·58-s − 1.60·64-s + 71-s + 1.94·73-s + 2.44·74-s + 1.80·76-s − 0.445·79-s − 1.56·83-s − 3.04·86-s − 1.80·89-s + 1.56·98-s − 1.24·101-s + 0.867·103-s + 1.80·109-s + 2.60·116-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

\[L(1/2,\rho) \approx 4.049159499\] \[L(1,\rho) \approx 2.451984715\]

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