Properties

Degree $2$
Conductor $10$
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 + 1.08·3-s + 0.5·4-s + 0.447·5-s − 0.764·6-s + 1.53·7-s − 0.353·8-s + 0.168·9-s − 0.316·10-s + 0.915·11-s + 0.540·12-s − 0.806·13-s − 1.08·14-s + 0.483·15-s + 0.250·16-s − 0.603·17-s − 0.118·18-s + 0.890·19-s + 0.223·20-s + 1.65·21-s − 0.647·22-s − 1.21·23-s − 0.382·24-s + 0.199·25-s + 0.570·26-s − 0.899·27-s + 0.765·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.64i)) \, \Gamma_{\R}(s+(1 - 4.64i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 10,\ (1 + 4.64274919042i, 1 - 4.64274919042i:\ ),\ 1)\)

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