LMFDB - L-function 4-588e2-1.1-c5e2-0-8

Dirichlet series

L(s) = 1

+ 9·3^-s − 34·5^-s + 332·11^-s + 2.05e3·13^-s − 306·15^-s + 922·17^-s + 452·19^-s + 3.77e3·23^-s + 3.12e3·25^-s − 729·27^-s + 2.33e3·29^-s − 9.79e3·31^-s + 2.98e3·33^-s − 2.39e3·37^-s + 1.84e4·39^-s + 1.44e4·41^-s + 9.30e3·43^-s + 2.46e4·47^-s + 8.29e3·51^-s − 1.11e3·53^-s − 1.12e4·55^-s + 4.06e3·57^-s + 4.68e4·59^-s − 9.76e3·61^-s − 6.97e4·65^-s + 2.62e4·67^-s + 3.39e4·69^-s + ⋯

L(s) = 1

+ 0.577·3^-s − 0.608·5^-s + 0.827·11^-s + 3.36·13^-s − 0.351·15^-s + 0.773·17^-s + 0.287·19^-s + 1.48·23^-s + 25^-s − 0.192·27^-s + 0.514·29^-s − 1.83·31^-s + 0.477·33^-s − 0.287·37^-s + 1.94·39^-s + 1.34·41^-s + 0.767·43^-s + 1.62·47^-s + 0.446·51^-s − 0.0542·53^-s − 0.503·55^-s + 0.165·57^-s + 1.75·59^-s − 0.335·61^-s − 2.04·65^-s + 0.714·67^-s + 0.859·69^-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}

\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree:	$4$
Conductor:	$345744$ = $2^{4} \cdot 3^{2} \cdot 7^{4}$
Sign:	$1$
Analytic conductor:	$8893.56$
Root analytic conductor:	$9.71111$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 345744,\ (\ :5/2, 5/2),\ 1)$

Particular Values

$L(3)$	$\approx$	$7.516932953$
$L(\frac12)$	$\approx$	$7.516932953$
$L(\frac{7}{2})$		not available
$L(1)$		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	$C_2$	$1 - p^{2} T + p^{4} T^{2}$
	7		$1$
good	5	$C_2^2$	$1 + 34 T - 1969 T^{2} + 34 p^{5} T^{3} + p^{10} T^{4}$
	11	$C_2^2$	$1 - 332 T - 50827 T^{2} - 332 p^{5} T^{3} + p^{10} T^{4}$
	13	$C_2$	$( 1 - 1026 T + p^{5} T^{2} )^{2}$
	17	$C_2^2$	$1 - 922 T - 569773 T^{2} - 922 p^{5} T^{3} + p^{10} T^{4}$
	19	$C_2^2$	$1 - 452 T - 2271795 T^{2} - 452 p^{5} T^{3} + p^{10} T^{4}$
	23	$C_2^2$	$1 - 3776 T + 7821833 T^{2} - 3776 p^{5} T^{3} + p^{10} T^{4}$
	29	$C_2$	$( 1 - 1166 T + p^{5} T^{2} )^{2}$
	31	$C_2^2$	$1 + 9792 T + 67254113 T^{2} + 9792 p^{5} T^{3} + p^{10} T^{4}$
	37	$C_2^2$	$1 + 2390 T - 63631857 T^{2} + 2390 p^{5} T^{3} + p^{10} T^{4}$
	41	$C_2$	$( 1 - 7230 T + p^{5} T^{2} )^{2}$
	43	$C_2$	$( 1 - 4652 T + p^{5} T^{2} )^{2}$
	47	$C_2^2$	$1 - 24672 T + 379362577 T^{2} - 24672 p^{5} T^{3} + p^{10} T^{4}$
	53	$C_2^2$	$1 + 1110 T - 416963393 T^{2} + 1110 p^{5} T^{3} + p^{10} T^{4}$
	59	$C_2^2$	$1 - 46892 T + 1483935365 T^{2} - 46892 p^{5} T^{3} + p^{10} T^{4}$
	61	$C_2^2$	$1 + 9762 T - 749299657 T^{2} + 9762 p^{5} T^{3} + p^{10} T^{4}$
	67	$C_2^2$	$1 - 26252 T - 660957603 T^{2} - 26252 p^{5} T^{3} + p^{10} T^{4}$
	71	$C_2$	$( 1 - 65440 T + p^{5} T^{2} )^{2}$
	73	$C_2^2$	$1 + 5606 T - 2041644357 T^{2} + 5606 p^{5} T^{3} + p^{10} T^{4}$
	79	$C_2^2$	$1 - 9840 T - 2980230799 T^{2} - 9840 p^{5} T^{3} + p^{10} T^{4}$
	83	$C_2$	$( 1 + 61108 T + p^{5} T^{2} )^{2}$
	89	$C_2^2$	$1 + 62958 T - 1620349685 T^{2} + 62958 p^{5} T^{3} + p^{10} T^{4}$
	97	$C_2$	$( 1 - 37838 T + p^{5} T^{2} )^{2}$
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L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.20416110408918513718029922972, −9.434824891745122419984065294869, −9.126141365531566457855009907065, −8.778074570163057877507852833064, −8.498396727634016014894141175242, −8.096891387053096270548866263471, −7.43608506073343121737789303452, −7.10274141324486456222338863923, −6.53475247485694311099989551014, −6.12644637395723930757046707358, −5.54117874671356423701372230269, −5.22168825446719444274542194233, −4.20894398792025486511339071699, −3.84632088306085466294044263234, −3.60211630900326925667652027265, −3.09821921420994812021126176586, −2.38595868765742204590771505893, −1.44321178541878259114200827147, −0.995619255154840588672015310130, −0.75944980829321898402086288270, 0.75944980829321898402086288270, 0.995619255154840588672015310130, 1.44321178541878259114200827147, 2.38595868765742204590771505893, 3.09821921420994812021126176586, 3.60211630900326925667652027265, 3.84632088306085466294044263234, 4.20894398792025486511339071699, 5.22168825446719444274542194233, 5.54117874671356423701372230269, 6.12644637395723930757046707358, 6.53475247485694311099989551014, 7.10274141324486456222338863923, 7.43608506073343121737789303452, 8.096891387053096270548866263471, 8.498396727634016014894141175242, 8.778074570163057877507852833064, 9.126141365531566457855009907065, 9.434824891745122419984065294869, 10.20416110408918513718029922972

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Properties

Origins

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Dirichlet series

Functional equation

Invariants

Particular Values

Euler product

Imaginary part of the first few zeros on the critical line

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

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4-588e2-1.1-c5e2-0-8
Degree	$4$
Conductor	$345744$
Sign	$1$
Analytic cond.	$8893.56$
Root an. cond.	$9.71111$
Motivic weight	$5$
Arithmetic	yes
Rational	yes
Primitive	no
Self-dual	yes
Analytic rank	$0$

Properties

Origins

Origins of factors

Downloads

Learn more

Particular Values

Imaginary part of the first few zeros on the critical line

Graph of the ZZZ-function along the critical line

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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