L-function 4-1-1.1-r0e4-m1.14p7.61p17.31m23.78-0

• Label: 4-1-1.1-r0e4-m1.14p7.61p17.31m23.78-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $2.02260$
• Root an. cond.: $1.19255$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.677 − 0.839i)2^-s + (−0.0660 − 0.190i)3^-s + (−0.217 − 1.13i)4^-s + (−0.722 + 0.445i)5^-s + (−0.204 − 0.0732i)6^-s + (0.161 − 0.260i)7^-s + (−0.404 + 0.227i)8^-s + (−0.335 + 0.0251i)9^-s + (−0.115 + 0.908i)10^-s + (0.155 − 2.12i)11^-s + (−0.201 + 0.116i)12^-s + (1.19 + 0.708i)13^-s + (−0.108 − 0.311i)14^-s + (0.132 + 0.107i)15^-s + (0.0739 + 0.461i)16^-s + (0.433 + 0.112i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-23.7i) \, \Gamma_{\R}(s-1.14i) \, \Gamma_{\R}(s+7.60i) \, \Gamma_{\R}(s+17.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$2.02260$
Root analytic conductor:	$1.19255$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (-23.7804846144i, -1.141485214492i, 7.60977421348i, 17.31219561542i:\ ),\ 1)$

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

−23.39664408, −22.56444137, −20.58138783, −15.40137552, −12.61591034, −4.29441687, 3.69633164, 5.94476917, 8.41565166, 10.97961379, 11.39737080, 13.52308045, 14.44034919, 16.38697267, 18.70071824, 19.66452990, 21.37405546, 23.35958806

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