L-function 4-336e2-1.1-c5e2-0-9

• Label: 4-336e2-1.1-c5e2-0-9
• 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 − 6·5^-s + 98·7^-s + 243·9^-s + 90·11^-s + 768·13^-s − 108·15^-s + 1.92e3·17^-s − 2.24e3·19^-s + 1.76e3·21^-s − 6.35e3·23^-s − 654·25^-s + 2.91e3·27^-s + 1.05e4·29^-s + 3.31e3·31^-s + 1.62e3·33^-s − 588·35^-s + 2.10e3·37^-s + 1.38e4·39^-s + 1.26e3·41^-s + 5.76e3·43^-s − 1.45e3·45^-s − 1.56e4·47^-s + 7.20e3·49^-s + 3.46e4·51^-s + 1.65e4·53^-s − 540·55^-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

$L(3)$	$\approx$	$6.991669387$
$L(\frac{7}{2})$		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	$D_{4}$	$1 + 6 T + 138 p T^{2} + 6 p^{5} T^{3} + p^{10} T^{4}$
	11	$D_{4}$	$1 - 90 T + 51246 T^{2} - 90 p^{5} T^{3} + p^{10} T^{4}$
	13	$D_{4}$	$1 - 768 T + 689558 T^{2} - 768 p^{5} T^{3} + p^{10} T^{4}$
	17	$D_{4}$	$1 - 1926 T + 3761514 T^{2} - 1926 p^{5} T^{3} + p^{10} T^{4}$
	19	$D_{4}$	$1 + 2248 T + 3007830 T^{2} + 2248 p^{5} T^{3} + p^{10} T^{4}$
	23	$D_{4}$	$1 + 6354 T + 22960446 T^{2} + 6354 p^{5} T^{3} + p^{10} T^{4}$
	29	$D_{4}$	$1 - 10572 T + 60053694 T^{2} - 10572 p^{5} T^{3} + p^{10} T^{4}$
	31	$D_{4}$	$1 - 3312 T + 31130942 T^{2} - 3312 p^{5} T^{3} + p^{10} T^{4}$
	37	$D_{4}$	$1 - 2104 T + 14492118 T^{2} - 2104 p^{5} T^{3} + p^{10} T^{4}$

Imaginary part of the first few zeros on the critical line

−10.73788599517428886758987050178, −10.44147114544742815034583253928, −9.852175836271121160601702220569, −9.838091991585157595872018825899, −8.714421872354062741163639907927, −8.579527667719131918440022482136, −8.283174240860898497130290318137, −7.77020517252972137557275260862, −7.44018738515136511795267339551, −6.47757871611835141016767793675, −6.20552494357699353835144655441, −5.72192341657863177969015414653, −4.62550956661090358858451257153, −4.50468082853022188551645698945, −3.57446630670126067340443351805, −3.51442178027130394397653100223, −2.42105519681349864822015371746, −2.03332029023314144094760677985, −1.25130220709984621670630565175, −0.68576481482159037703123855094, 0.68576481482159037703123855094, 1.25130220709984621670630565175, 2.03332029023314144094760677985, 2.42105519681349864822015371746, 3.51442178027130394397653100223, 3.57446630670126067340443351805, 4.50468082853022188551645698945, 4.62550956661090358858451257153, 5.72192341657863177969015414653, 6.20552494357699353835144655441, 6.47757871611835141016767793675, 7.44018738515136511795267339551, 7.77020517252972137557275260862, 8.283174240860898497130290318137, 8.579527667719131918440022482136, 8.714421872354062741163639907927, 9.838091991585157595872018825899, 9.852175836271121160601702220569, 10.44147114544742815034583253928, 10.73788599517428886758987050178

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