L-function 4-1-1.1-r0e4-m3.29p3.44p21.02m21.17-0

• Label: 4-1-1.1-r0e4-m3.29p3.44p21.02m21.17-0
• Degree: $4$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $3.21079$
• Root an. cond.: $1.33860$
• Arithmetic: no
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (1.25 + 0.746i)2^-s + (0.310 − 1.45i)3^-s + (0.400 + 1.87i)4^-s + (−0.0767 + 0.0627i)5^-s + (1.47 − 1.59i)6^-s + (−0.540 + 0.326i)7^-s + (−0.419 + 1.44i)8^-s + (−0.898 − 0.906i)9^-s + (−0.143 + 0.0215i)10^-s + (−0.382 − 0.203i)11^-s + (2.85 − 0.00108i)12^-s + (−0.0268 + 0.182i)13^-s + (−0.922 + 0.00613i)14^-s + (0.0677 + 0.131i)15^-s + (−0.718 + 0.339i)16^-s + (0.361 + 0.291i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-21.1i) \, \Gamma_{\R}(s-3.29i) \, \Gamma_{\R}(s+3.44i) \, \Gamma_{\R}(s+21.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}$

Invariants

Degree:	$4$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$3.21079$
Root analytic conductor:	$1.33860$
Rational:	no
Arithmetic:	no
Primitive:	yes
Self-dual:	no
Selberg data:	$(4,\ 1,\ (-21.1709752712i, -3.29346016214i, 3.44054733534i, 21.023888098i:\ ),\ 1)$

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

−23.5760440, −22.1860928, −19.8074830, −15.9923107, −14.6742523, −13.2972617, −10.8924504, −9.8163386, −4.9323140, 6.5620540, 8.0103796, 12.4318524, 12.9878376, 14.4237577, 16.4912885, 18.5813021, 22.8439508, 24.2069230

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