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 + 1.59·3-s + 0.5·4-s − 1.49·5-s − 1.12·6-s − 0.366·7-s − 0.353·8-s + 1.53·9-s + 1.06·10-s + 0.856·11-s + 0.795·12-s − 0.530·13-s + 0.259·14-s − 2.38·15-s + 0.250·16-s + 1.04·17-s − 1.08·18-s + 1.18·19-s − 0.749·20-s − 0.583·21-s − 0.605·22-s − 0.370·23-s − 0.562·24-s + 1.24·25-s + 0.375·26-s + 0.850·27-s − 0.183·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 2 ^{s/2} \, \Gamma_{\R}(s+25.0i) \, \Gamma_{\R}(s-25.0i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 2,\ (25.0524237851i, -25.0524237851i:\ ),\ 1)\)

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