Properties

Degree $2$
Conductor $41$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive no
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 2·2-s + 3·4-s + 2·5-s + 4·8-s + 9-s + 4·10-s + 5·16-s + 2·18-s + 6·20-s + 2·23-s + 3·25-s + 2·31-s + 6·32-s + 3·36-s + 2·37-s + 8·40-s + 41-s + 2·43-s + 2·45-s + 4·46-s + 49-s + 6·50-s + 2·59-s + 2·61-s + 4·62-s + 7·64-s + 4·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 41,\ (0, 0:\ ),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -1.617129879\]
Pole at \(s=1\)

Euler product

\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{41}(40, \cdot))\)

Imaginary part of the first few zeros on the critical line

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