Properties

Degree $2$
Conductor $31$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.159·2-s − 0.347·3-s − 0.974·4-s + 1.76·5-s − 0.0553·6-s + 0.716·7-s − 0.314·8-s − 0.879·9-s + 0.281·10-s + 1.27·11-s + 0.338·12-s − 0.554·13-s + 0.114·14-s − 0.615·15-s + 0.924·16-s − 1.31·17-s − 0.139·18-s + 0.306·19-s − 1.72·20-s − 0.249·21-s + 0.202·22-s + 0.0352·23-s + 0.109·24-s + 2.12·25-s − 0.0882·26-s + 0.653·27-s − 0.698·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1.75i) \, \Gamma_{\R}(s-1.75i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 31,\ (1.7546428498i, -1.7546428498i:\ ),\ 1)\)

Euler product

\(L(s,f) = \displaystyle\prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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