Properties

Degree 2
Conductor 5
Sign $0.804 + 0.593i$
Self-dual no
Motivic weight 5

Origins

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Normalization:  

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Dirichlet series

$L(s,f)$  = 1  + 1.172i ·2-s − 1.276i·3-s − 0.375·4-s + (−0.804 + 0.593i) 5-s + 1.496·6-s − 0.460i·7-s + 0.732i ·8-s − 0.629·9-s + (−0.695 − 0.943i) 10-s + 0.627·11-s + 0.478i ·12-s − 0.195i·13-s + 0.539·14-s + (0.757 + 1.027i) 15-s − 1.234·16-s − 0.578i·17-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut & 5 ^{s/2} \Gamma_{\C}(s+2.5) \cdot L(s,f)\cr =\mathstrut & (0.804 + 0.593i) \Lambda(1-s,\overline{f}) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $0.804 + 0.593i$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(2,\ 5,\ (\ :5/2),\ 0.804 + 0.593i)$

Euler product

\[\begin{equation} L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1} \end{equation}\]

Particular Values

\[L(1/2,f) \approx 0.8783964683 + 0.2887276942i\] \[L(1,f) \approx 0.9290785379 + 0.2533076847i\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line