Properties

Degree 2
Conductor $ 2 \cdot 5 $
Sign $1$
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s − 0.487·3-s + 0.5·4-s − 0.447·5-s − 0.344·6-s + 1.51·7-s + 0.353·8-s − 0.762·9-s − 0.316·10-s − 0.791·11-s − 0.243·12-s − 0.112·13-s + 1.07·14-s + 0.217·15-s + 0.250·16-s + 0.408·17-s − 0.539·18-s + 0.736·19-s − 0.223·20-s − 0.739·21-s − 0.559·22-s − 0.317·23-s − 0.172·24-s + 0.199·25-s − 0.0793·26-s + 0.858·27-s + 0.758·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+(1 + 2.55i)) \, \Gamma_{\R}(s+(1 - 2.55i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 10,\ (1 + 2.55636946085i, 1 - 2.55636946085i:\ ),\ 1)\)

Euler product

\[\begin{aligned}L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line