L-function 4-1-1.1-r0e4-p1.20m3.82m19.88p22.50-0

• Label: 4-1-1.1-r0e4-p1.20m3.82m19.88p22.50-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $1.18063$
• Root an. cond.: $1.04238$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.513 − 0.833i)2^-s + (−0.0450 + 1.33i)3^-s + (0.0506 + 0.855i)4^-s + (−0.0405 − 0.440i)5^-s + (1.13 − 0.645i)6^-s + (−0.317 + 0.0815i)7^-s + (−0.0737 − 0.0491i)8^-s + (−0.363 − 0.119i)9^-s + (−0.345 + 0.259i)10^-s + (0.986 + 0.239i)11^-s + (−1.14 + 0.0289i)12^-s + (−0.298 − 0.973i)13^-s + (0.230 + 0.222i)14^-s + (0.587 − 0.0341i)15^-s + (−0.0210 + 0.498i)16^-s + (−0.151 + 0.392i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.4i) \, \Gamma_{\R}(s+1.20i) \, \Gamma_{\R}(s-3.81i) \, \Gamma_{\R}(s-19.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$1.18063$
Root analytic conductor:	$1.04238$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (22.49640718i, 1.201562854542i, -3.81917738186i, -19.8787926526i:\ ),\ 1)$

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

−24.954738, −24.180181, −19.252675, −18.443401, −16.793458, −15.064627, −13.455939, −11.663928, −9.323828, −7.150648, −6.535784, 9.030178, 10.510768, 12.277778, 15.316804, 17.035571, 20.847349, 22.365493, 24.915810

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