Properties

Degree $2$
Conductor $15$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.951·2-s − 0.577·3-s − 0.0951·4-s − 0.447·5-s + 0.549·6-s + 0.0716·7-s + 1.04·8-s + 0.333·9-s + 0.425·10-s + 0.219·11-s + 0.0549·12-s − 0.589·13-s − 0.0681·14-s + 0.258·15-s − 0.895·16-s + 1.83·17-s − 0.317·18-s − 0.694·19-s + 0.0425·20-s − 0.0413·21-s − 0.209·22-s + 1.60·23-s − 0.601·24-s + 0.199·25-s + 0.560·26-s − 0.192·27-s − 0.00681·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s+4.75i) \, \Gamma_{\R}(s-4.75i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 15,\ (4.75479713395i, -4.75479713395i:\ ),\ 1)\)

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