L-function 4-1-1.1-r0e4-p1.95m4.74m18.21p21.00-0

• Label: 4-1-1.1-r0e4-p1.95m4.74m18.21p21.00-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $2.22489$
• Root an. cond.: $1.22131$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.553 + 0.741i)2^-s + (−0.00758 + 0.0796i)3^-s + (−0.0630 + 0.820i)4^-s + (−0.564 − 0.0667i)5^-s + (−0.0632 + 0.0384i)6^-s + (1.33 + 0.273i)7^-s + (0.00997 − 0.200i)8^-s + (−0.504 − 0.00120i)9^-s + (−0.263 − 0.455i)10^-s + (−0.578 − 0.551i)11^-s + (−0.0648 − 0.0112i)12^-s + (0.0593 + 0.523i)13^-s + (0.534 + 1.13i)14^-s + (0.00959 − 0.0444i)15^-s + (−0.00150 + 0.0438i)16^-s + (−0.108 − 0.782i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.0i) \, \Gamma_{\R}(s+1.95i) \, \Gamma_{\R}(s-4.74i) \, \Gamma_{\R}(s-18.2i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$2.22489$
Root analytic conductor:	$1.22131$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (21.0013177272i, 1.95152402253i, -4.74207909766i, -18.21076265204i:\ ),\ 1)$

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

−24.22162400, −23.04708087, −20.02707661, −17.71380605, −15.15629091, −13.74444211, −11.75536751, −10.69204911, −7.97514359, −4.91033784, 7.97811213, 11.64270161, 14.04564600, 15.98081824, 20.98395063, 22.63590375, 24.06263678, 24.98876898

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