L-function 4-1-1.1-r0e4-m1.78p6.26p18.87m23.35-0

• Label: 4-1-1.1-r0e4-m1.78p6.26p18.87m23.35-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $3.06472$
• Root an. cond.: $1.32311$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−1.18 − 0.259i)2^-s + (−0.937 − 0.452i)3^-s + (0.944 + 0.612i)4^-s + (−0.0116 + 0.184i)5^-s + (0.989 + 0.777i)6^-s + (0.202 + 0.342i)7^-s + (−1.68 − 0.609i)8^-s + (0.802 + 0.849i)9^-s + (0.0616 − 0.214i)10^-s + (0.165 + 0.313i)11^-s + (−0.608 − 1.00i)12^-s + (−0.293 − 0.195i)13^-s + (−0.150 − 0.457i)14^-s + (0.0944 − 0.167i)15^-s + (1.93 + 0.923i)16^-s + (−0.155 + 0.605i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-23.3i) \, \Gamma_{\R}(s-1.78i) \, \Gamma_{\R}(s+6.25i) \, \Gamma_{\R}(s+18.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$3.06472$
Root analytic conductor:	$1.32311$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (-23.3467915216i, -1.78026796581i, 6.25689320186i, 18.87016628548i:\ ),\ 1)$

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

−24.69052130, −23.55793043, −21.29614712, −17.36143583, −15.53523224, −11.57051808, −9.59172868, −0.43794484, 6.20232148, 8.01521872, 9.87251034, 11.46460632, 12.51530713, 15.31800018, 17.07002502, 17.97993013, 19.12128902, 21.30368127, 24.86004591

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