Properties

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

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.707·2-s − 0.764·3-s + 0.5·4-s − 0.759·5-s + 0.540·6-s − 1.23·7-s − 0.353·8-s − 0.415·9-s + 0.537·10-s − 1.30·11-s − 0.382·12-s − 0.417·13-s + 0.874·14-s + 0.581·15-s + 0.250·16-s + 1.91·17-s + 0.293·18-s + 0.491·19-s − 0.379·20-s + 0.945·21-s + 0.920·22-s − 0.931·23-s + 0.270·24-s − 0.422·25-s + 0.295·26-s + 1.08·27-s − 0.618·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 2 ^{s/2} \, \Gamma_{\R}(s+19.1i) \, \Gamma_{\R}(s-19.1i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 2,\ (19.1254224082i, -19.1254224082i:\ ),\ 1)\)

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