Properties

Label 3-2e2-1.1-r0e3-m2.64m8.24p10.88-0
Degree $3$
Conductor $4$
Sign $1$
Analytic cond. $3.78867$
Root an. cond. $1.55893$
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Dirichlet series

L(s)  = 1  +(0.5)·2-s + (1.06 − 0.481i)3-s +(0.25)·4-s + (−0.248 + 0.0721i)5-s + (0.533 − 0.240i)6-s + (0.676 + 0.291i)7-s +(0.125)·8-s + (−0.161 − 1.50i)9-s + (−0.124 + 0.0360i)10-s + (−0.264 + 0.294i)11-s + (0.266 − 0.120i)12-s + (−0.771 + 0.00362i)13-s + (0.338 + 0.145i)14-s + (−0.230 + 0.196i)15-s +(0.0625)·16-s + (0.335 + 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s-8.23i) \, \Gamma_{\R}(s-2.64i) \, \Gamma_{\R}(s+10.8i) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Analytic conductor: \(3.78867\)
Root analytic conductor: \(1.55893\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: \((3,\ 4,\ (-8.239796i, -2.641226i, 10.881024i:\ ),\ 1)\)

Euler product

\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.1266, −22.6026, −21.3171, −20.2680, −19.1905, −16.8507, −15.0792, −13.8334, −8.0717, −4.6967, −2.6576, 12.2527, 14.2232, 15.1645, 17.5728, 19.1542, 20.3333, 21.3176, 23.0757, 24.1020

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