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  − 1.06·2-s − 1.20·3-s + 0.124·4-s + 1.42·5-s + 1.27·6-s − 0.909·7-s + 0.928·8-s + 0.441·9-s − 1.51·10-s + 0.105·11-s − 0.148·12-s + 0.388·13-s + 0.964·14-s − 1.71·15-s − 1.10·16-s − 0.369·17-s − 0.468·18-s + 0.736·19-s + 0.176·20-s + 1.09·21-s − 0.112·22-s + 0.741·23-s − 1.11·24-s + 1.02·25-s − 0.412·26-s + 0.670·27-s − 0.112·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1.01i) \, \Gamma_{\R}(s-1.01i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 29,\ (1.0172665508i, -1.0172665508i:\ ),\ 1)\)

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