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 − 0.869·3-s + 0.5·4-s − 0.447·5-s − 0.614·6-s − 0.574·7-s + 0.353·8-s − 0.243·9-s − 0.316·10-s − 1.89·11-s − 0.434·12-s − 0.595·13-s − 0.406·14-s + 0.388·15-s + 0.250·16-s − 1.26·17-s − 0.172·18-s + 0.725·19-s − 0.223·20-s + 0.499·21-s − 1.33·22-s − 1.52·23-s − 0.307·24-s + 0.200·25-s − 0.421·26-s + 1.08·27-s − 0.287·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+(1 + 23.2i)) \, \Gamma_{\R}(s+(1 - 23.2i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 10,\ (1 + 23.246400292i, 1 - 23.246400292i:\ ),\ 1)$

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