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 − 0.0984·3-s + 0.5·4-s + 0.447·5-s + 0.0696·6-s − 1.24·7-s − 0.353·8-s − 0.990·9-s − 0.316·10-s − 1.16·11-s − 0.0492·12-s + 0.371·13-s + 0.880·14-s − 0.0440·15-s + 0.250·16-s + 0.387·17-s + 0.700·18-s + 1.09·19-s + 0.223·20-s + 0.122·21-s + 0.825·22-s + 0.896·23-s + 0.0348·24-s + 0.200·25-s − 0.262·26-s + 0.195·27-s − 0.622·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+25.2i) \, \Gamma_{\R}(s-25.2i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 10,\ (25.2347442706i, -25.2347442706i:\ ),\ -1)$

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