Properties

Degree $1$
Conductor $20627$
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  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20627 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(20627\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((1,\ 20627,\ (1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 4.216652320\] \[L(1,\rho) \approx 1.290576788\]

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line