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  + 0.590·2-s + 0.388·3-s − 0.650·4-s − 0.879·5-s + 0.229·6-s + 1.67·7-s − 0.975·8-s − 0.849·9-s − 0.519·10-s − 0.0888·11-s − 0.252·12-s + 1.08·13-s + 0.992·14-s − 0.341·15-s + 0.0745·16-s + 0.797·17-s − 0.501·18-s − 0.757·19-s + 0.572·20-s + 0.652·21-s − 0.0524·22-s + 0.494·23-s − 0.378·24-s − 0.227·25-s + 0.639·26-s − 0.718·27-s − 1.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.06i)) \, \Gamma_{\R}(s+(1 - 1.06i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 31,\ (1 + 1.06284037124i, 1 - 1.06284037124i:\ ),\ 1)\)

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