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  − 0.769·2-s + 1.78·3-s − 0.408·4-s − 0.677·5-s − 1.36·6-s + 0.495·7-s + 1.08·8-s + 2.17·9-s + 0.520·10-s − 0.958·11-s − 0.727·12-s + 1.51·13-s − 0.380·14-s − 1.20·15-s − 0.424·16-s − 0.848·17-s − 1.67·18-s + 0.221·19-s + 0.276·20-s + 0.881·21-s + 0.737·22-s − 0.600·23-s + 1.92·24-s − 0.541·25-s − 1.16·26-s + 2.08·27-s − 0.202·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1.83i) \, \Gamma_{\R}(s-1.83i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 31,\ (1.83128343514i, -1.83128343514i:\ ),\ 1)\)

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