Properties

Label 4-1-1.1-r0e4-p1.85m4.51m19.59p22.25-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $2.27924$
Root an. cond. $1.22870$
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Dirichlet series

L(s)  = 1  + (0.674 + 0.590i)2-s + (−0.382 + 0.156i)3-s + (0.551 + 0.796i)4-s + (−0.535 + 0.125i)5-s + (−0.350 − 0.119i)6-s + (0.151 − 0.120i)7-s + (0.876 + 0.535i)8-s + (0.569 − 0.119i)9-s + (−0.435 − 0.231i)10-s + (−0.602 − 0.457i)11-s + (−0.335 − 0.217i)12-s + (−0.602 − 0.00348i)13-s + (0.173 + 0.00785i)14-s + (0.184 − 0.131i)15-s + (0.324 + 1.23i)16-s + (0.290 + 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.2i) \, \Gamma_{\R}(s+1.85i) \, \Gamma_{\R}(s-4.50i) \, \Gamma_{\R}(s-19.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(2.27924\)
Root analytic conductor: \(1.22870\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: \((4,\ 1,\ (22.2500458752i, 1.852039992048i, -4.50889918758i, -19.59318667958i:\ ),\ 1)\)

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

−23.39273141, −20.44072297, −18.83244589, −16.48950296, −14.87732783, −13.00557715, −11.66655489, −10.32196893, −7.35350569, −4.90148759, 7.39147020, 11.01853893, 13.02429540, 15.24744890, 16.66008845, 21.76685994, 23.30359953, 24.48310810

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