Properties

Degree $4$
Conductor $2624$
Sign $1$
Motivic weight $1$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,f)$  = 1  − 2-s − 0.221·7-s + 8-s + 1.60·9-s + 0.221·14-s − 16-s − 0.200·17-s − 1.60·18-s − 0.589·23-s − 1.13·25-s − 1.73·31-s + 0.200·34-s + 0.414·41-s + 0.589·46-s + 1.15·47-s − 0.571·49-s + 1.13·50-s − 0.221·56-s + 1.73·62-s − 0.356·63-s + 64-s − 0.879·71-s + 1.60·72-s − 1.55·73-s + 0.384·79-s + 1.59·81-s − 0.414·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\[L(1/2,f) \approx 0.4396912148\] \[L(1,f) \approx 0.5921530565\]

Euler product

\(L(s,f) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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