Properties

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

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 1.17·2-s − 0.947·3-s + 0.371·4-s − 1.03·5-s − 1.10·6-s + 1.14·7-s − 0.735·8-s − 0.101·9-s − 1.21·10-s − 0.382·11-s − 0.352·12-s + 1.47·13-s + 1.33·14-s + 0.980·15-s − 1.23·16-s + 1.32·17-s − 0.119·18-s + 1.01·19-s − 0.384·20-s − 1.08·21-s − 0.448·22-s − 0.355·23-s + 0.697·24-s + 0.0698·25-s + 1.72·26-s + 1.04·27-s + 0.424·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+34.0i) \, \Gamma_{\R}(s-34.0i) \, L(s,f)\cr=\mathstrut & \,\Lambda(1-s,f)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 1,\ (34.0278842001i, -34.0278842001i:\ ),\ 1)\)

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