L-function 4-294e2-1.1-c5e2-0-6

• Label: 4-294e2-1.1-c5e2-0-6
• Degree: $4$
• Conductor: $86436$
• Sign: $1$
• Analytic cond.: $2223.39$
• Root an. cond.: $6.86679$
• Motivic weight: $5$
• Arithmetic: yes
• Rational: yes
• Primitive: no
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 4·2^-s − 9·3^-s + 86·5^-s − 36·6^-s − 64·8^-s + 344·10^-s − 34·11^-s + 6·13^-s − 774·15^-s − 256·16^-s − 1.90e3·17^-s − 1.48e3·19^-s − 136·22^-s + 224·23^-s + 576·24^-s + 3.12e3·25^-s + 24·26^-s + 729·27^-s − 1.30e4·29^-s − 3.09e3·30^-s + 1.73e3·31^-s + 306·33^-s − 7.61e3·34^-s + 7.63e3·37^-s − 5.95e3·38^-s − 54·39^-s − 5.50e3·40^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}$

Invariants

Degree:	$4$
Conductor:	$86436$    =    $2^{2} \cdot 3^{2} \cdot 7^{4}$
Sign:	$1$
Analytic conductor:	$2223.39$
Root analytic conductor:	$6.86679$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 86436,\ (\ :5/2, 5/2),\ 1)$

Particular Values

$L(3)$	$\approx$	$1.367228021$
$L(\frac{7}{2})$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2	$C_2$	$1 - p^{2} T + p^{4} T^{2}$
	3	$C_2$	$1 + p^{2} T + p^{4} T^{2}$
	7		$1$
good	5	$C_2^2$	$1 - 86 T + 4271 T^{2} - 86 p^{5} T^{3} + p^{10} T^{4}$
	11	$C_2^2$	$1 + 34 T - 159895 T^{2} + 34 p^{5} T^{3} + p^{10} T^{4}$
	13	$C_2$	$( 1 - 3 T + p^{5} T^{2} )^{2}$
	17	$C_2^2$	$1 + 112 p T + 7631 p^{2} T^{2} + 112 p^{6} T^{3} + p^{10} T^{4}$
	19	$C_2^2$	$1 + 1489 T - 258978 T^{2} + 1489 p^{5} T^{3} + p^{10} T^{4}$
	23	$C_2^2$	$1 - 224 T - 6386167 T^{2} - 224 p^{5} T^{3} + p^{10} T^{4}$
	29	$C_2$	$( 1 + 6508 T + p^{5} T^{2} )^{2}$
	31	$C_2^2$	$1 - 1731 T - 25632790 T^{2} - 1731 p^{5} T^{3} + p^{10} T^{4}$
	37	$C_2^2$	$1 - 7633 T - 11081268 T^{2} - 7633 p^{5} T^{3} + p^{10} T^{4}$

Imaginary part of the first few zeros on the critical line

−11.27387736229650752234088444444, −10.71931116464294747510896777831, −10.35552324586099047025922338445, −9.768045403887808591763810122435, −9.306327293045087413871770529446, −8.774284993239319606535949538733, −8.682455427172272337749185291269, −7.44498123599887760610706494109, −7.29495143853881468371535156304, −6.43624624542158109095244120233, −5.93230521568187760191337984999, −5.91022029342992991163260945346, −5.23193735000298371912898166415, −4.56765105067927125306035083518, −4.19065861651640633201959575238, −3.39851478785159545803861195973, −2.41966581545510576503914260785, −2.14641973567871489123447945075, −1.40329686489076269739106779520, −0.27647657844965387612131433877, 0.27647657844965387612131433877, 1.40329686489076269739106779520, 2.14641973567871489123447945075, 2.41966581545510576503914260785, 3.39851478785159545803861195973, 4.19065861651640633201959575238, 4.56765105067927125306035083518, 5.23193735000298371912898166415, 5.91022029342992991163260945346, 5.93230521568187760191337984999, 6.43624624542158109095244120233, 7.29495143853881468371535156304, 7.44498123599887760610706494109, 8.682455427172272337749185291269, 8.774284993239319606535949538733, 9.306327293045087413871770529446, 9.768045403887808591763810122435, 10.35552324586099047025922338445, 10.71931116464294747510896777831, 11.27387736229650752234088444444

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