Properties

Degree 4
Conductor $ 5^{2} \cdot 109 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes

Related objects

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Normalization:  

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Dirichlet series

$L(s,f)$  = 1  − 0.618·4-s + 0.552·5-s + 0.951·9-s − 1.99·11-s − 0.618·16-s + 1.05·19-s − 0.341·20-s − 0.694·25-s − 0.502·29-s − 0.359·31-s − 0.587·36-s − 0.482·41-s + 1.23·44-s + 0.525·45-s + 0.727·49-s − 1.10·55-s − 0.631·59-s + 0.949·61-s + 64-s + 0.293·71-s − 0.654·76-s + 1.81·79-s − 0.341·80-s − 0.0949·81-s + 2.40·89-s + 0.585·95-s − 1.89·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 2725 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2725\)    =    \(5^{2} \cdot 109\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2725,\ (\ :1/2, 1/2),\ 1)$

Euler product

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

Particular Values

\[L(1/2,f) \approx 0.6661068253\] \[L(1,f) \approx 0.8669723303\]

Imaginary part of the first few zeros on the critical line

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