Properties

Degree 4
Conductor $ 5^{2} \cdot 131 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s,f)$  = 1  − 0.866·4-s + 0.774·5-s + 0.488·9-s − 0.139·11-s − 0.250·16-s − 0.917·19-s − 0.670·20-s − 0.400·25-s − 1.57·29-s + 1.73·31-s − 0.422·36-s − 0.727·41-s + 0.121·44-s + 0.378·45-s − 1.77·49-s − 0.108·55-s + 0.104·59-s + 0.699·61-s + 1.08·64-s + 2.59·71-s + 0.794·76-s + 1.50·79-s − 0.193·80-s − 0.761·81-s − 0.0983·89-s − 0.710·95-s − 0.0682·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 3275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3275\)    =    \(5^{2} \cdot 131\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3275,\ (\ :1/2, 1/2),\ 1)$

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

\[L(1/2,f) \approx 0.7028852954\] \[L(1,f) \approx 0.8838301864\]

Imaginary part of the first few zeros on the critical line

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