Properties

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

Related objects

Learn more about

Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s − 0.853·3-s + 0.5·4-s + 0.447·5-s − 0.603·6-s + 1.78·7-s + 0.353·8-s − 0.270·9-s + 0.316·10-s − 1.40·11-s − 0.426·12-s − 0.452·13-s + 1.26·14-s − 0.381·15-s + 0.250·16-s + 1.43·17-s − 0.191·18-s + 0.0535·19-s + 0.223·20-s − 1.52·21-s − 0.996·22-s + 1.53·23-s − 0.301·24-s + 0.199·25-s − 0.319·26-s + 1.08·27-s + 0.894·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+12.4i) \, \Gamma_{\R}(s-12.4i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 10,\ (12.4596489912i, -12.4596489912i:\ ),\ 1)\)

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