Properties

Label 4-1-1.1-r0e4-c9.57c17.99-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $18.9899$
Root an. cond. $2.08752$
Arithmetic no
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

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Dirichlet series

L(s)  = 1  − 3.02·2-s − 0.0417·3-s + 4.92·4-s + 1.29·5-s + 0.126·6-s + 0.175·7-s − 5.20·8-s + 1.91·9-s − 3.91·10-s + 1.48·11-s − 0.205·12-s − 0.105·13-s − 0.531·14-s − 0.0541·15-s + 3.14·16-s + 2.10·17-s − 5.78·18-s + 0.182·19-s + 6.39·20-s − 0.00733·21-s − 4.49·22-s + 1.82·23-s + 0.217·24-s + 0.0441·25-s + 0.320·26-s − 0.201·27-s + 0.866·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.9i) \, \Gamma_{\R}(s+9.56i) \, \Gamma_{\R}(s-17.9i) \, \Gamma_{\R}(s-9.56i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(18.9899\)
Root analytic conductor: \(2.08752\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 1,\ (17.9921157465i, 9.56738695728i, -17.9921157465i, -9.56738695728i:\ ),\ 1)\)

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

−24.97671444, −21.22693602, −18.94514608, −16.87801311, −9.80360970, −7.11008907, −1.37888092, −1.31463971, 1.31463971, 1.37888092, 7.11008907, 9.80360970, 16.87801311, 18.94514608, 21.22693602, 24.97671444

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