L-function 4-1-1.1-r0e4-p1.63m4.25m19.47p22.09-0

• Label: 4-1-1.1-r0e4-p1.63m4.25m19.47p22.09-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $1.83941$
• Root an. cond.: $1.16458$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.0231 + 0.218i)2^-s + (0.725 + 0.511i)3^-s + (−1.24 − 0.0101i)4^-s + (0.459 + 0.798i)5^-s + (−0.128 + 0.146i)6^-s + (1.08 − 0.509i)7^-s + (0.0356 − 0.751i)8^-s + (−0.547 + 0.742i)9^-s + (−0.184 + 0.0819i)10^-s + (0.254 + 0.0145i)11^-s + (−0.898 − 0.644i)12^-s + (0.0585 − 0.494i)13^-s + (0.0862 + 0.247i)14^-s + (−0.0749 + 0.814i)15^-s + (0.704 + 0.0373i)16^-s + (0.117 + 0.897i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.0i) \, \Gamma_{\R}(s+1.62i) \, \Gamma_{\R}(s-4.25i) \, \Gamma_{\R}(s-19.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$1.83941$
Root analytic conductor:	$1.16458$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (22.0947457086i, 1.628025115328i, -4.25436733726i, -19.4684034867i:\ ),\ 1)$

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

−24.00378304, −20.46928041, −18.43867388, −17.31901118, −14.43542522, −13.61952832, −11.93300114, −9.12479214, −8.42792671, −5.04136528, 8.46570807, 10.34402271, 13.74282677, 14.67734871, 17.60702206, 21.68550847, 23.07879845

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