L-function 4-1-1.1-r0e4-m1.97p5.10p15.91m19.04-0

• Label: 4-1-1.1-r0e4-m1.97p5.10p15.91m19.04-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $1.91798$
• Root an. cond.: $1.17682$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.340 − 0.484i)2^-s + (0.0264 − 0.577i)3^-s + (−0.790 − 0.329i)4^-s + (1.16 + 0.105i)5^-s + (−0.270 − 0.209i)6^-s + (0.165 − 0.0772i)7^-s + (−0.316 + 1.08i)8^-s + (−0.536 − 0.0305i)9^-s + (0.447 − 0.528i)10^-s + (−0.442 − 0.761i)11^-s + (−0.211 + 0.448i)12^-s + (0.756 + 0.0103i)13^-s + (0.0188 − 0.106i)14^-s + (0.0917 − 0.671i)15^-s + (0.296 + 0.742i)16^-s + (−0.200 + 0.739i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-19.0i) \, \Gamma_{\R}(s-1.97i) \, \Gamma_{\R}(s+5.10i) \, \Gamma_{\R}(s+15.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$1.91798$
Root analytic conductor:	$1.17682$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (-19.03767623312i, -1.972435011632i, 5.10355437536i, 15.9065568694i:\ ),\ 1)$

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

−23.20477758, −22.06186511, −20.90486526, −17.99614909, −13.51432196, −9.56782573, 5.68994723, 8.76421280, 10.91443624, 13.20797304, 14.09560366, 17.62720365, 21.16901722, 22.76927297, 24.08379404

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