Properties

Degree $2$
Conductor $31$
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 + 2·7-s + 4·8-s + 9-s + 4·10-s + 4·14-s + 5·16-s + 2·18-s + 2·19-s + 6·20-s + 3·25-s + 6·28-s + 31-s + 6·32-s + 4·35-s + 3·36-s + 4·38-s + 8·40-s + 2·41-s + 2·45-s + 2·47-s + 3·49-s + 6·50-s + 8·56-s + 2·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((2,\ 31,\ (\ :0),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -3.225385518\]
Pole at \(s=1\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{31}(30, \cdot))\)

Imaginary part of the first few zeros on the critical line

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