Properties

Degree 4
Conductor $ 5^{2} \cdot 61 $
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  − 1.61·4-s + 1.27·5-s − 0.460·9-s − 1.16·11-s + 1.61·16-s − 1.71·19-s − 2.06·20-s + 0.629·25-s + 2.10·29-s + 0.924·31-s + 0.745·36-s − 0.119·41-s + 1.88·44-s − 0.587·45-s + 0.638·49-s − 1.48·55-s − 0.111·59-s − 0.128·61-s − 1.00·64-s + 0.0453·71-s + 2.77·76-s − 0.364·79-s + 2.06·80-s − 0.787·81-s − 0.0964·89-s − 2.18·95-s + 0.535·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1525\)    =    \(5^{2} \cdot 61\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((4,\ 1525,\ (\ :1/2, 1/2),\ 1)\)

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

\[L(1/2,f) \approx 0.5065018928\] \[L(1,f) \approx 0.7423837223\]

Imaginary part of the first few zeros on the critical line

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