Properties

Degree 4
Conductor $ 1 $
Sign $1$
Primitive yes
Self-dual yes
Analytic rank 0

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Dirichlet series

$L(s,f)$  = 1  + 1.34·2-s − 0.187·3-s + 0.464·4-s − 0.00162·5-s − 0.251·6-s + 0.228·7-s + 0.169·8-s − 0.463·9-s − 0.00218·10-s + 0.695·11-s − 0.0870·12-s − 0.882·13-s + 0.306·14-s + 0.000304·15-s + 0.408·16-s + 0.716·17-s − 0.622·18-s − 0.927·19-s − 0.000699·20-s − 0.0427·21-s + 0.934·22-s + 0.419·23-s − 0.0309·24-s + 0.227·25-s − 1.17·26-s + 0.100·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+12.4i) \, \Gamma_{\R}(s+4.72i) \, \Gamma_{\R}(s-12.4i) \, \Gamma_{\R}(s-4.72i) \, L(s,f)\cr =\mathstrut & \,\Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 1,\ (12.46875226152i, 4.72095103638i, -12.46875226152i, -4.72095103638i:\ ),\ 1)$

Euler product

\[\begin{aligned} L(s,f) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.342525613, −23.108966361, −22.396069285, −21.193386862, −19.439354578, −17.114451933, −14.496061510, 14.496061510, 17.114451933, 19.439354578, 21.193386862, 22.396069285, 23.108966361, 24.342525613

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