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  − 1.82·2-s + 1.17·3-s + 2.33·4-s + 0.882·5-s − 2.13·6-s + 0.0850·7-s − 2.43·8-s + 0.372·9-s − 1.61·10-s + 0.479·11-s + 2.73·12-s − 0.480·13-s − 0.155·14-s + 1.03·15-s + 2.11·16-s − 1.88·17-s − 0.680·18-s + 0.570·19-s + 2.06·20-s + 0.0995·21-s − 0.876·22-s − 0.282·23-s − 2.85·24-s − 0.220·25-s + 0.877·26-s − 0.735·27-s + 0.198·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.34i)) \, \Gamma_{\R}(s+(1 - 1.34i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 31,\ (1 + 1.343010459i, 1 - 1.343010459i:\ ),\ 1)\)

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