L-function 4-1-1.1-r0e4-p1.69m4.66m16.94p19.92-0

• Label: 4-1-1.1-r0e4-p1.69m4.66m16.94p19.92-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $1.64447$
• Root an. cond.: $1.13241$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.0582 − 0.197i)2^-s + (−0.382 + 1.42i)3^-s + (0.573 − 0.0229i)4^-s + (−0.129 − 0.576i)5^-s + (0.259 + 0.158i)6^-s + (−0.00648 + 0.436i)7^-s + (0.122 − 0.0372i)8^-s + (−0.664 − 1.09i)9^-s + (−0.121 − 0.00794i)10^-s + (0.308 + 0.472i)11^-s + (−0.186 + 0.826i)12^-s + (−0.238 − 0.170i)13^-s + (0.0858 + 0.0267i)14^-s + (0.871 + 0.0350i)15^-s + (−0.609 − 0.0403i)16^-s + (0.792 − 0.305i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+19.9i) \, \Gamma_{\R}(s+1.68i) \, \Gamma_{\R}(s-4.66i) \, \Gamma_{\R}(s-16.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$1.64447$
Root analytic conductor:	$1.13241$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (19.91852255416i, 1.685122031456i, -4.66217749248i, -16.94146709314i:\ ),\ 1)$

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

−24.888105826, −24.118722660, −22.623360609, −18.570212028, −16.531522798, −14.144390623, −12.483164499, −10.906249272, −7.605965466, −6.485730757, 9.696292750, 11.751809942, 15.511258986, 20.319738275, 21.360091279, 22.737784517, 24.456683572

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