Properties

Degree $2$
Conductor $6$
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 − 0.577·3-s + 0.5·4-s − 1.36·5-s + 0.408·6-s + 0.182·7-s − 0.353·8-s + 0.333·9-s + 0.961·10-s − 1.24·11-s − 0.288·12-s − 0.133·13-s − 0.128·14-s + 0.785·15-s + 0.250·16-s + 0.990·17-s − 0.235·18-s − 1.23·19-s − 0.680·20-s − 0.105·21-s + 0.879·22-s − 0.551·23-s + 0.204·24-s + 0.849·25-s + 0.0944·26-s − 0.192·27-s + 0.0912·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 6 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.54i)) \, \Gamma_{\R}(s+(1 - 4.54i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 6,\ (1 + 4.54845492142i, 1 - 4.54845492142i:\ ),\ -1)\)

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