Properties

Label 4-1-1.1-r0e4-c8.54c20.41-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $19.4702$
Root an. cond. $2.10059$
Arithmetic no
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

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Dirichlet series

L(s)  = 1  + 0.202·2-s − 0.717·3-s − 0.738·4-s − 0.616·5-s − 0.145·6-s − 0.242·7-s − 0.104·8-s − 0.767·9-s − 0.124·10-s − 0.378·11-s + 0.530·12-s + 1.79·13-s − 0.0491·14-s + 0.442·15-s − 0.404·16-s + 0.573·17-s − 0.155·18-s + 1.66·19-s + 0.455·20-s + 0.174·21-s − 0.0766·22-s + 0.407·23-s + 0.0752·24-s − 1.15·25-s + 0.363·26-s + 0.754·27-s + 0.179·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+20.4i) \, \Gamma_{\R}(s+8.54i) \, \Gamma_{\R}(s-20.4i) \, \Gamma_{\R}(s-8.54i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(19.4702\)
Root analytic conductor: \(2.10059\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 1,\ (20.405089661i, 8.54276551352i, -20.405089661i, -8.54276551352i:\ ),\ 1)\)

Euler product

\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.4215999, −22.7568773, −18.1015439, −16.0658283, −13.6493218, −11.4956074, −5.5265424, −3.5846613, −0.6608225, 0.6608225, 3.5846613, 5.5265424, 11.4956074, 13.6493218, 16.0658283, 18.1015439, 22.7568773, 23.4215999

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