Properties

Degree 4
Conductor $ 2^{6} \cdot 71 $
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  − 1.22·2-s + 0.499·4-s + 1.03·7-s + 0.612·8-s − 0.666·9-s − 1.26·14-s − 1.25·16-s + 1.26·17-s + 0.816·18-s + 1.77·23-s − 1.40·25-s + 0.516·28-s + 0.0962·31-s + 0.918·32-s − 1.54·34-s − 0.333·36-s − 1.00·41-s − 2.17·46-s + 0.572·47-s − 1.02·49-s + 1.71·50-s + 0.632·56-s − 0.117·62-s − 0.688·63-s + 0.125·64-s + 0.630·68-s − 1.03·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 4544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4544\)    =    \(2^{6} \cdot 71\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 4544,\ (\ :1/2, 1/2),\ 1)$

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

\[L(1/2,f) \approx 0.4817769348\] \[L(1,f) \approx 0.5904593145\]

Imaginary part of the first few zeros on the critical line

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