L-function 4-336e2-1.1-c5e2-0-8

• Label: 4-336e2-1.1-c5e2-0-8
• Degree: $4$
• Conductor: $112896$
• Sign: $1$
• Analytic cond.: $2904.02$
• Root an. cond.: $7.34091$
• Motivic weight: $5$
• Arithmetic: yes
• Rational: yes
• Primitive: no
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 18·3^-s − 64·5^-s − 98·7^-s + 243·9^-s − 540·11^-s + 204·13^-s − 1.15e3·15^-s + 304·17^-s + 1.12e3·19^-s − 1.76e3·21^-s + 940·23^-s − 2.18e3·25^-s + 2.91e3·27^-s + 932·29^-s + 1.64e4·31^-s − 9.72e3·33^-s + 6.27e3·35^-s − 1.76e3·37^-s + 3.67e3·39^-s − 2.55e3·41^-s + 2.46e4·43^-s − 1.55e4·45^-s + 3.67e4·47^-s + 7.20e3·49^-s + 5.47e3·51^-s + 9.16e3·53^-s + 3.45e4·55^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}$

Invariants

Degree:	$4$
Conductor:	$112896$    =    $2^{8} \cdot 3^{2} \cdot 7^{2}$
Sign:	$1$
Analytic conductor:	$2904.02$
Root analytic conductor:	$7.34091$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 112896,\ (\ :5/2, 5/2),\ 1)$

Particular Values

$L(3)$	$\approx$	$3.864606427$
$L(\frac{7}{2})$		not available

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2		$1$
	3	$C_1$	$( 1 - p^{2} T )^{2}$
	7	$C_1$	$( 1 + p^{2} T )^{2}$
good	5	$D_{4}$	$1 + 64 T + 6278 T^{2} + 64 p^{5} T^{3} + p^{10} T^{4}$
	11	$D_{4}$	$1 + 540 T + 170902 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4}$
	13	$D_{4}$	$1 - 204 T + 498014 T^{2} - 204 p^{5} T^{3} + p^{10} T^{4}$
	17	$D_{4}$	$1 - 304 T + 272222 T^{2} - 304 p^{5} T^{3} + p^{10} T^{4}$
	19	$D_{4}$	$1 - 1120 T - 188298 T^{2} - 1120 p^{5} T^{3} + p^{10} T^{4}$
	23	$D_{4}$	$1 - 940 T + 8072750 T^{2} - 940 p^{5} T^{3} + p^{10} T^{4}$
	29	$D_{4}$	$1 - 932 T + 41139854 T^{2} - 932 p^{5} T^{3} + p^{10} T^{4}$
	31	$D_{4}$	$1 - 16408 T + 122456382 T^{2} - 16408 p^{5} T^{3} + p^{10} T^{4}$
	37	$D_{4}$	$1 + 1764 T + 117649454 T^{2} + 1764 p^{5} T^{3} + p^{10} T^{4}$

Imaginary part of the first few zeros on the critical line

−10.76369386288637598233115183051, −10.48108282958140639165345696746, −9.796871795521251433747920260483, −9.792923732357535811659437980389, −8.877903386482260819738267938439, −8.643817319318985614139412850841, −7.921472774005890649044041785065, −7.902888211080406742560822979675, −7.22335309741655018471536186417, −6.97915552497176790829687037549, −5.90921688628840550378988581801, −5.80387098553843759783705087714, −4.58345863832954192183181489964, −4.52589599392285324292687284254, −3.47167453273810960455547794605, −3.46449914107610976806436296690, −2.50478223405585195365999899732, −2.35678817938169513331655673666, −0.873912671042865415000643638330, −0.63614273554842050961409129086, 0.63614273554842050961409129086, 0.873912671042865415000643638330, 2.35678817938169513331655673666, 2.50478223405585195365999899732, 3.46449914107610976806436296690, 3.47167453273810960455547794605, 4.52589599392285324292687284254, 4.58345863832954192183181489964, 5.80387098553843759783705087714, 5.90921688628840550378988581801, 6.97915552497176790829687037549, 7.22335309741655018471536186417, 7.902888211080406742560822979675, 7.921472774005890649044041785065, 8.643817319318985614139412850841, 8.877903386482260819738267938439, 9.792923732357535811659437980389, 9.796871795521251433747920260483, 10.48108282958140639165345696746, 10.76369386288637598233115183051

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