L-function 4-1-1.1-r0e4-p1.97m5.10m15.91p19.04-0

• Label: 4-1-1.1-r0e4-p1.97m5.10m15.91p19.04-0
• 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

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

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