Properties

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

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,f)$  = 1  − 0.301·11-s − 16-s − 1.97·31-s + 1.40·41-s − 0.128·61-s + 2.25·71-s − 81-s + 2.88·101-s + 0.909·121-s − 0.961·131-s + 0.325·151-s + 0.301·176-s − 0.817·181-s − 2.96·191-s − 0.0688·211-s − 1.03·241-s + 0.252·251-s + 256-s − 1.88·271-s − 0.656·281-s + 2.77·311-s − 3.35·331-s + 0.595·341-s − 2·361-s + 1.44·401-s + 0.926·421-s − 1.73·431-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3125\)    =    \(5^{5}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3125,\ (\ :1/2, 1/2),\ 1)$

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

\[L(1/2,f) \approx 0.7183136284\] \[L(1,f) \approx 0.9084063132\]

Imaginary part of the first few zeros on the critical line

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