Properties

Label 4-1-1.1-r0e4-m2.22p4.47p21.65m23.90-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $3.25166$
Root an. cond. $1.34284$
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Dirichlet series

L(s)  = 1  + (0.233 − 1.56i)2-s + (1.02 − 0.267i)3-s + (−0.709 − 0.728i)4-s + (−0.457 + 1.28i)5-s + (−0.178 − 1.66i)6-s + (−0.306 + 0.0628i)7-s + (−0.679 − 0.116i)8-s + (0.624 − 0.550i)9-s + (1.89 + 1.01i)10-s + (0.172 + 0.233i)11-s + (−0.924 − 0.559i)12-s + (−0.997 − 0.459i)13-s + (0.0266 + 0.493i)14-s + (−0.127 + 1.43i)15-s + (−0.0350 − 0.186i)16-s + (−0.196 − 0.0185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.6i) \, \Gamma_{\R}(s+4.46i) \, \Gamma_{\R}(s-2.22i) \, \Gamma_{\R}(s-23.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(3.25166\)
Root analytic conductor: \(1.34284\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: \((4,\ 1,\ (21.6515998894i, 4.46755486328i, -2.22332030312i, -23.8958344496i:\ ),\ 1)\)

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

−24.86118025, −24.27912543, −19.87483120, −16.63990326, −15.94864304, −14.48040072, −12.68031797, −8.95555088, −7.49541914, 3.04158682, 7.28818408, 9.75538726, 11.04390092, 12.34948425, 14.14675135, 15.20440903, 18.40989391, 19.60274792, 20.19321749, 22.04354699

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