L-function 4-1-1.1-r0e4-m0.06m7.93m16.63p24.63-0

• Label: 4-1-1.1-r0e4-m0.06m7.93m16.63p24.63-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $0.0646165$
• Root an. cond.: $0.504180$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−1.91 + 0.0377i)2^-s + (−0.799 − 0.0323i)3^-s + (1.08 − 0.144i)4^-s + (−0.606 + 0.961i)5^-s + (1.52 + 0.0317i)6^-s + (−0.242 + 0.0323i)7^-s + (0.916 + 0.181i)8^-s + (−0.134 + 0.0517i)9^-s + (1.12 − 1.85i)10^-s + (−0.976 − 0.0482i)11^-s + (−0.872 + 0.0799i)12^-s + (−0.137 + 0.249i)13^-s + (0.462 − 0.0709i)14^-s + (0.516 − 0.748i)15^-s + (−1.88 + 0.0562i)16^-s + (−0.292 + 0.852i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+24.6i) \, \Gamma_{\R}(s-0.0621i) \, \Gamma_{\R}(s-7.93i) \, \Gamma_{\R}(s-16.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$0.0646165$
Root analytic conductor:	$0.504180$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (24.625120245i, -0.062191724021i, -7.93038878332i, -16.63253973772i:\ ),\ 1)$

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

−22.8650342, −20.3642624, −18.9905560, −17.8583767, −16.9152521, −15.9554273, −13.0949839, −11.4224256, −9.9565801, −8.7739335, −7.6539659, −4.9978665, −0.3963276, 10.6322979, 18.1098191, 19.3345203, 22.3645305, 23.5455547

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