L-function 4-810e2-1.1-c3e2-0-20

• Label: 4-810e2-1.1-c3e2-0-20
• Degree: $4$
• Conductor: $656100$
• Sign: $1$
• Analytic cond.: $2284.03$
• Root an. cond.: $6.91314$
• Motivic weight: $3$
• Arithmetic: yes
• Rational: yes
• Primitive: no
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 2·2^-s − 5·5^-s + 4·7^-s − 8·8^-s − 10·10^-s − 48·11^-s − 2·13^-s + 8·14^-s − 16·16^-s + 228·17^-s + 280·19^-s − 96·22^-s + 72·23^-s − 4·26^-s + 210·29^-s − 272·31^-s + 456·34^-s − 20·35^-s − 668·37^-s + 560·38^-s + 40·40^-s − 198·41^-s + 268·43^-s + 144·46^-s + 216·47^-s + 343·49^-s + 156·53^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}$

Invariants

Degree:	$4$
Conductor:	$656100$    =    $2^{2} \cdot 3^{8} \cdot 5^{2}$
Sign:	$1$
Analytic conductor:	$2284.03$
Root analytic conductor:	$6.91314$
Motivic weight:	$3$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 656100,\ (\ :3/2, 3/2),\ 1)$

Particular Values

$L(2)$	$\approx$	$4.870273790$
$L(\frac{5}{2})$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2	$C_2$	$1 - p T + p^{2} T^{2}$
	3		$1$
	5	$C_2$	$1 + p T + p^{2} T^{2}$
good	7	$C_2^2$	$1 - 4 T - 327 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}$
	11	$C_2^2$	$1 + 48 T + 973 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4}$
	13	$C_2^2$	$1 + 2 T - 2193 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}$
	17	$C_2$	$( 1 - 114 T + p^{3} T^{2} )^{2}$
	19	$C_2$	$( 1 - 140 T + p^{3} T^{2} )^{2}$
	23	$C_2^2$	$1 - 72 T - 6983 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4}$
	29	$C_2^2$	$1 - 210 T + 19711 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4}$
	31	$C_2^2$	$1 + 272 T + 44193 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4}$
	37	$C_2$	$( 1 + 334 T + p^{3} T^{2} )^{2}$

Imaginary part of the first few zeros on the critical line

−10.15654237602487405573337490213, −9.880599521252069768472748533219, −9.042345667320217032631065944274, −8.994855848758337873062899141920, −8.224241849301056216918337304864, −7.69031941439357464882675871816, −7.49682792104555461771904144127, −7.34696231272868379192233176262, −6.66862086101920496331719031501, −5.67861809605018058164066081574, −5.49414733823382487935011348593, −5.15419101051553368630762217472, −5.12196319973062219200578936599, −4.07213867728197129802241045458, −3.45053147291626498129051716096, −3.11130975132747923062393873450, −3.02986739073575827442992807115, −1.83626307391308889629290670794, −1.03960538447744075143387245361, −0.64242935669352938397950764797, 0.64242935669352938397950764797, 1.03960538447744075143387245361, 1.83626307391308889629290670794, 3.02986739073575827442992807115, 3.11130975132747923062393873450, 3.45053147291626498129051716096, 4.07213867728197129802241045458, 5.12196319973062219200578936599, 5.15419101051553368630762217472, 5.49414733823382487935011348593, 5.67861809605018058164066081574, 6.66862086101920496331719031501, 7.34696231272868379192233176262, 7.49682792104555461771904144127, 7.69031941439357464882675871816, 8.224241849301056216918337304864, 8.994855848758337873062899141920, 9.042345667320217032631065944274, 9.880599521252069768472748533219, 10.15654237602487405573337490213

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