L-function 4-588e2-1.1-c7e2-0-7

• Label: 4-588e2-1.1-c7e2-0-7
• Degree: $4$
• Conductor: $345744$
• Sign: $1$
• Analytic cond.: $33739.2$
• Root an. cond.: $13.5529$
• Motivic weight: $7$
• Arithmetic: yes
• Rational: yes
• Primitive: no
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 27·3^-s − 100·5^-s − 2.77e3·11^-s − 6.58e3·13^-s − 2.70e3·15^-s − 5.90e3·17^-s − 6.64e3·19^-s − 1.98e3·23^-s + 7.81e4·25^-s − 1.96e4·27^-s − 4.16e5·29^-s + 1.17e5·31^-s − 7.48e4·33^-s + 3.35e5·37^-s − 1.77e5·39^-s − 5.30e5·41^-s − 1.86e5·43^-s + 6.57e5·47^-s − 1.59e5·51^-s + 6.08e5·53^-s + 2.77e5·55^-s − 1.79e5·57^-s + 5.36e5·59^-s + 1.79e6·61^-s + 6.58e5·65^-s − 2.12e6·67^-s − 5.35e4·69^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}$

Invariants

Degree:	$4$
Conductor:	$345744$    =    $2^{4} \cdot 3^{2} \cdot 7^{4}$
Sign:	$1$
Analytic conductor:	$33739.2$
Root analytic conductor:	$13.5529$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 345744,\ (\ :7/2, 7/2),\ 1)$

Particular Values

$L(4)$	$\approx$	$2.804682592$
$L(\frac{9}{2})$		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	$C_2$	$1 - p^{3} T + p^{6} T^{2}$
	7		$1$
good	5	$C_2^2$	$1 + 4 p^{2} T - 109 p^{4} T^{2} + 4 p^{9} T^{3} + p^{14} T^{4}$
	11	$C_2^2$	$1 + 2774 T - 11792095 T^{2} + 2774 p^{7} T^{3} + p^{14} T^{4}$
	13	$C_2$	$( 1 + 3294 T + p^{7} T^{2} )^{2}$
	17	$C_2^2$	$1 + 5900 T - 375528673 T^{2} + 5900 p^{7} T^{3} + p^{14} T^{4}$
	19	$C_2^2$	$1 + 6644 T - 849729003 T^{2} + 6644 p^{7} T^{3} + p^{14} T^{4}$
	23	$C_2^2$	$1 + 1982 T - 3400897123 T^{2} + 1982 p^{7} T^{3} + p^{14} T^{4}$
	29	$C_2$	$( 1 + 208106 T + p^{7} T^{2} )^{2}$
	31	$C_2^2$	$1 - 117792 T - 13637658847 T^{2} - 117792 p^{7} T^{3} + p^{14} T^{4}$
	37	$C_2^2$	$1 - 335686 T + 17753213463 T^{2} - 335686 p^{7} T^{3} + p^{14} T^{4}$

Imaginary part of the first few zeros on the critical line

−9.526310789008828306810841051286, −9.495818987264873884522411299940, −8.785263276893203329811772806620, −8.616739772922722157718291234629, −7.81551376058665930169679895286, −7.79158005181949934800826407394, −7.05102207197775877800884172710, −7.02072238307116102495631711165, −6.06285218555691671597483061057, −5.80497468872726357241131205572, −4.96147929170139513697898724779, −4.95749975630065742813192181952, −4.02602473199963266099005443967, −3.82208079131516615931526356892, −3.02111947435136813503268888911, −2.75950054500881910774087486123, −1.92356497833029215926701013168, −1.85958056382846429834314100631, −0.67074930057982823709831660355, −0.41759672478793209281249335355, 0.41759672478793209281249335355, 0.67074930057982823709831660355, 1.85958056382846429834314100631, 1.92356497833029215926701013168, 2.75950054500881910774087486123, 3.02111947435136813503268888911, 3.82208079131516615931526356892, 4.02602473199963266099005443967, 4.95749975630065742813192181952, 4.96147929170139513697898724779, 5.80497468872726357241131205572, 6.06285218555691671597483061057, 7.02072238307116102495631711165, 7.05102207197775877800884172710, 7.79158005181949934800826407394, 7.81551376058665930169679895286, 8.616739772922722157718291234629, 8.785263276893203329811772806620, 9.495818987264873884522411299940, 9.526310789008828306810841051286

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