Properties

Label 3-1-1.1-r0e3-p0.17p16.40m16.57-0
Degree $3$
Conductor $1$
Sign $1$
Analytic cond. $0.0488655$
Root an. cond. $0.365595$
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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This L-function has the smallest analytic conductor among degree 3 L-functions with signature (0,0,0;).

Dirichlet series

L(s)  = 1  + (−0.421 + 1.06i)2-s + (−0.768 − 1.31i)3-s + (−0.541 + 0.167i)4-s + (−0.400 + 0.239i)5-s + (1.72 − 0.266i)6-s + (−0.117 + 0.553i)7-s + (−0.268 − 0.648i)8-s + (−0.366 + 0.703i)9-s + (−0.0872 − 0.528i)10-s + (−0.0411 − 0.100i)11-s + (0.635 + 0.582i)12-s + (−0.309 − 0.328i)13-s + (−0.541 − 0.358i)14-s + (0.622 + 0.341i)15-s + (−0.0226 + 0.546i)16-s + (0.259 − 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+16.4i) \, \Gamma_{\R}(s+0.171i) \, \Gamma_{\R}(s-16.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(0.0488655\)
Root analytic conductor: \(0.365595\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: \((3,\ 1,\ (16.403124740291375i, 0.17112189172831185i, -16.574246632019687i:\ ),\ 1)\)

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

−23.4129437373605, −21.6868858848525, −19.8646960903853, −11.1407921358216, −9.8664332915609, −4.6144521141879, 6.4222353306131, 7.8655276953699, 12.3429586556897, 18.3920792539766, 22.6875461169111, 24.2616362328567

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