Properties

Label 3-1-1.1-r0e3-p0.11p30.23m30.34-0
Degree $3$
Conductor $1$
Sign $1$
Analytic cond. $0.131415$
Root an. cond. $0.508411$
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Dirichlet series

L(s)  = 1  + (−0.915 − 0.588i)2-s + (−0.446 + 1.40i)3-s + (1.40 + 0.489i)4-s + (−0.268 + 0.269i)5-s + (1.23 − 1.02i)6-s + (0.837 + 0.244i)7-s + (−1.18 − 1.27i)8-s + (−1.31 + 0.149i)9-s + (0.404 − 0.0887i)10-s + (−0.173 + 0.512i)11-s + (−1.31 + 1.75i)12-s + (−0.735 − 1.19i)13-s + (−0.622 − 0.716i)14-s + (−0.257 − 0.496i)15-s + (0.997 + 0.899i)16-s + (−0.303 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+30.2i) \, \Gamma_{\R}(s+0.113i) \, \Gamma_{\R}(s-30.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(0.131415\)
Root analytic conductor: \(0.508411\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: \((3,\ 1,\ (30.2296585152i, 0.11322826136i, -30.3428867764i:\ ),\ 1)\)

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

−24.25733139, −24.03173748, −20.56426801, −19.02543825, −17.57097756, −16.63089370, −14.62687920, −12.20137658, −11.21848270, −8.40982311, −7.26222562, −6.15734969, −1.76067050, 2.65834675, 5.10842017, 7.63221024, 9.77383326, 10.82641519, 11.64242094, 15.14915869, 16.08171901, 17.64474396, 19.74352396, 20.82935662, 22.23442041

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