L-function 8-832e4-1.1-c3e4-0-3

• Label: 8-832e4-1.1-c3e4-0-3
• Degree: $8$
• Conductor: $479174066176$
• Sign: $1$
• Analytic cond.: $5.80707\times 10^{6}$
• Root an. cond.: $7.00639$
• Motivic weight: $3$
• Arithmetic: yes
• Rational: yes
• Primitive: no
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 108·9^-s + 500·25^-s − 1.24e3·53^-s + 3.52e3·61^-s + 7.29e3·81^-s + 5.99e3·101^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 4.39e3·169^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + 197^-s + 199^-s + 211^-s + 223^-s − 5.40e4·225^-s + 227^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}$

Invariants

Degree:	$8$
Conductor:	$2^{24} \cdot 13^{4}$
Sign:	$1$
Analytic conductor:	$5.80707\times 10^{6}$
Root analytic conductor:	$7.00639$
Motivic weight:	$3$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$

Particular Values

$L(2)$	$\approx$	$3.667044665$
$L(\frac{5}{2})$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2		$1$
	13	$C_2$	$( 1 - p^{3} T^{2} )^{2}$
good	3	$C_2$	$( 1 + p^{3} T^{2} )^{4}$
	5	$C_2$	$( 1 - p^{3} T^{2} )^{4}$
	7	$C_2^2$$\times$$C_2^2$	$( 1 - 38 T + 722 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} )( 1 + 38 T + 722 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} )$
	11	$C_2^2$$\times$$C_2^2$	$( 1 - 90 T + 4050 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} )( 1 + 90 T + 4050 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} )$
	17	$C_2^2$	$( 1 + 9774 T^{2} + p^{6} T^{4} )^{2}$
	19	$C_2^2$$\times$$C_2^2$	$( 1 - 70 T + 2450 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} )( 1 + 70 T + 2450 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} )$
	23	$C_2$	$( 1 + p^{3} T^{2} )^{4}$
	29	$C_2^2$	$( 1 - 14922 T^{2} + p^{6} T^{4} )^{2}$
	31	$C_2^2$$\times$$C_2^2$	$( 1 - 70 T + 2450 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} )( 1 + 70 T + 2450 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} )$

Imaginary part of the first few zeros on the critical line

−6.74427609543950363013272719585, −6.66741823588049881352802126708, −6.54353147828337698666703729377, −6.19836069450057957923683370302, −6.00266617180120222714258715862, −5.71288533483853666480131884160, −5.53095773382860514302446923384, −5.22198392408249587632132871924, −5.14720708482754466018344338874, −4.89866130620108967124075190529, −4.71194650319374922516667788164, −4.42718308063220226979175489295, −3.83455120889396641225603986057, −3.63177432487925349151270965328, −3.25502484387917690549498134236, −3.16890763955198625754088585273, −3.10794551535238449424817721915, −2.55723349344410007001382285977, −2.37351055017806941999448819779, −2.28997265261417135209137563614, −1.82872458409628457585154922889, −1.13903369046564440223613104004, −0.68327313731946522732718797010, −0.66981311173468473817263537004, −0.34912008491548672885151133734, 0.34912008491548672885151133734, 0.66981311173468473817263537004, 0.68327313731946522732718797010, 1.13903369046564440223613104004, 1.82872458409628457585154922889, 2.28997265261417135209137563614, 2.37351055017806941999448819779, 2.55723349344410007001382285977, 3.10794551535238449424817721915, 3.16890763955198625754088585273, 3.25502484387917690549498134236, 3.63177432487925349151270965328, 3.83455120889396641225603986057, 4.42718308063220226979175489295, 4.71194650319374922516667788164, 4.89866130620108967124075190529, 5.14720708482754466018344338874, 5.22198392408249587632132871924, 5.53095773382860514302446923384, 5.71288533483853666480131884160, 6.00266617180120222714258715862, 6.19836069450057957923683370302, 6.54353147828337698666703729377, 6.66741823588049881352802126708, 6.74427609543950363013272719585

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