Properties

Degree $2$
Conductor $29$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.0427·2-s + 1.12·3-s − 0.998·4-s + 0.790·5-s + 0.0479·6-s − 1.03·7-s − 0.0854·8-s + 0.258·9-s + 0.0337·10-s − 0.892·11-s − 1.11·12-s + 1.55·13-s − 0.0443·14-s + 0.886·15-s + 0.994·16-s + 1.43·17-s + 0.0110·18-s − 0.887·19-s − 0.788·20-s − 1.16·21-s − 0.0381·22-s + 0.470·23-s − 0.0958·24-s − 0.375·25-s + 0.0665·26-s − 0.831·27-s + 1.03·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.37i)) \, \Gamma_{\R}(s+(1 - 1.37i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 29,\ (1 + 1.37929245765i, 1 - 1.37929245765i:\ ),\ 1)\)

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