L-function 4-1-1.1-r0e4-p1.49m4.26m18.43p21.20-0

• Label: 4-1-1.1-r0e4-p1.49m4.26m18.43p21.20-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $1.51070$
• Root an. cond.: $1.10865$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.191 − 0.389i)2^-s + (0.0834 + 1.28i)3^-s + (−0.375 + 0.149i)4^-s + (0.557 + 0.0152i)5^-s + (0.486 − 0.279i)6^-s + (−0.307 − 0.580i)7^-s + (−0.0116 + 0.608i)8^-s + (−0.0815 + 0.215i)9^-s + (−0.100 − 0.220i)10^-s + (−0.514 − 0.676i)11^-s + (−0.223 − 0.471i)12^-s + (0.718 + 0.942i)13^-s + (−0.167 + 0.230i)14^-s + (0.0268 + 0.719i)15^-s + (−0.474 − 0.150i)16^-s + (0.690 − 0.132i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.2i) \, \Gamma_{\R}(s+1.49i) \, \Gamma_{\R}(s-4.26i) \, \Gamma_{\R}(s-18.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$1.51070$
Root analytic conductor:	$1.10865$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (21.2021329724i, 1.490163268882i, -4.2609146966i, -18.43138154474i:\ ),\ 1)$

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

−24.94483471, −23.28270845, −18.69489724, −17.86686196, −15.55370998, −13.45103740, −12.52574938, −9.75443634, −7.76611026, −6.12063354, 9.30431542, 10.77704704, 13.75368079, 16.16324808, 20.85816701, 21.83756915, 23.60388982

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