Properties

Label 4-1-1.1-r0e4-c5.56c17.39-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $5.97805$
Root an. cond. $1.56365$
Arithmetic no
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

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Dirichlet series

L(s)  = 1  + 0.938·2-s + 0.906·3-s + 0.105·4-s + 1.59·5-s + 0.850·6-s − 0.756·7-s + 0.310·8-s − 0.422·9-s + 1.49·10-s − 0.142·11-s + 0.0957·12-s − 0.875·13-s − 0.709·14-s + 1.44·15-s + 0.0897·16-s − 0.566·17-s − 0.396·18-s + 0.916·19-s + 0.168·20-s − 0.685·21-s − 0.134·22-s − 0.148·23-s + 0.281·24-s + 1.10·25-s − 0.821·26-s − 0.604·27-s − 0.0798·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.3i) \, \Gamma_{\R}(s+5.55i) \, \Gamma_{\R}(s-17.3i) \, \Gamma_{\R}(s-5.55i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(5.97805\)
Root analytic conductor: \(1.56365\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 1,\ (17.38767874744i, 5.5596400005i, -17.38767874744i, -5.5596400005i:\ ),\ 1)\)

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

−24.72274057, −22.71154679, −21.75768323, −20.08216834, −14.17824306, −13.26579338, −9.57307120, −2.52853940, 2.52853940, 9.57307120, 13.26579338, 14.17824306, 20.08216834, 21.75768323, 22.71154679, 24.72274057

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