Properties

Degree $1$
Conductor $7$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(7\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((1,\ 7,\ (1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 1.146585666\] \[L(1,\rho) \approx 1.187410411\]

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