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 + 1.40·3-s + 0.5·4-s + 0.447·5-s − 0.995·6-s − 0.886·7-s − 0.353·8-s + 0.981·9-s − 0.316·10-s − 0.419·11-s + 0.703·12-s − 1.64·13-s + 0.626·14-s + 0.629·15-s + 0.250·16-s + 1.69·17-s − 0.694·18-s + 1.54·19-s + 0.223·20-s − 1.24·21-s + 0.296·22-s − 0.105·23-s − 0.497·24-s + 0.199·25-s + 1.16·26-s − 0.0259·27-s − 0.443·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+21.6i) \, \Gamma_{\R}(s-21.6i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 10,\ (21.6514464961i, -21.6514464961i:\ ),\ -1)\)

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