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 − 1.37·3-s + 0.5·4-s − 0.447·5-s + 0.969·6-s − 0.756·7-s − 0.353·8-s + 0.879·9-s + 0.316·10-s − 0.360·11-s − 0.685·12-s − 0.944·13-s + 0.534·14-s + 0.613·15-s + 0.250·16-s − 0.666·17-s − 0.621·18-s + 1.65·19-s − 0.223·20-s + 1.03·21-s + 0.254·22-s + 0.325·23-s + 0.484·24-s + 0.199·25-s + 0.667·26-s + 0.165·27-s − 0.378·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+(1 + 15.9i)) \, \Gamma_{\R}(s+(1 - 15.9i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 10,\ (1 + 15.9987139994i, 1 - 15.9987139994i:\ ),\ -1)\)

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