Properties

Degree 4
Conductor 775
Sign $1$
Self-dual yes
Motivic weight 1

Related objects

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Normalization:  

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Dirichlet series

$L(s,f)$  = 1  − 1.5·4-s − 0.894·5-s + 0.666·9-s + 1.250·16-s + 1.341·20-s − 0.200·25-s − 0.742·29-s + 1.257·31-s − 36-s − 1.874·41-s − 0.596·45-s + 0.285·49-s + 1.041·59-s + 0.512·61-s − 0.375·64-s − 0.949·71-s + 1.800·79-s − 1.118·80-s − 0.555·81-s + 0.423·89-s + 0.300·100-s − 0.398·101-s − 0.383·109-s + 1.114·116-s − 0.545·121-s − 1.885·124-s + 1.073·125-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut & 775 ^{s/2} \Gamma_{\C}(s+0.5) ^{2} \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(775\)    =    \(5^{2} \cdot 31\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 775,\ (\ :1/2, 1/2),\ 1)$

Euler product

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

Particular Values

\[L(1/2,f) \approx 0.3599289595\] \[L(1,f) \approx 0.6077041963\]

Imaginary part of the first few zeros on the critical line

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