LMFDB - L-function 8-1083e4-1.1-c5e4-0-0

Dirichlet series

L(s) = 1

− 2^-s + 36·3^-s − 37·4^-s − 8·5^-s − 36·6^-s − 142·7^-s + 97·8^-s + 810·9^-s + 8·10^-s − 714·11^-s − 1.33e3·12^-s + 74·13^-s + 142·14^-s − 288·15^-s − 509·16^-s − 3.69e3·17^-s − 810·18^-s + 296·20^-s − 5.11e3·21^-s + 714·22^-s + 862·23^-s + 3.49e3·24^-s − 4.57e3·25^-s − 74·26^-s + 1.45e4·27^-s + 5.25e3·28^-s − 992·29^-s + ⋯

L(s) = 1

− 0.176·2^-s + 2.30·3^-s − 1.15·4^-s − 0.143·5^-s − 0.408·6^-s − 1.09·7^-s + 0.535·8^-s + 10/3·9^-s + 0.0252·10^-s − 1.77·11^-s − 2.67·12^-s + 0.121·13^-s + 0.193·14^-s − 0.330·15^-s − 0.497·16^-s − 3.09·17^-s − 0.589·18^-s + 0.165·20^-s − 2.52·21^-s + 0.314·22^-s + 0.339·23^-s + 1.23·24^-s − 1.46·25^-s − 0.0214·26^-s + 3.84·27^-s + 1.26·28^-s − 0.219·29^-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree:	$8$
Conductor:	$3^{4} \cdot 19^{8}$
Sign:	$1$
Analytic conductor:	$9.10240\times 10^{8}$
Root analytic conductor:	$13.1793$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$4$
Selberg data:	$(8,\ 3^{4} \cdot 19^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )$

Particular Values

$L(3)$	$=$	$0$
$L(\frac12)$	$=$	$0$
$L(\frac{7}{2})$		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	3	$C_1$	$( 1 - p^{2} T )^{4}$
	19		$1$
good	2	$C_2 \wr S_4$	$1 + T + 19 p T^{2} - 11 p T^{3} + 449 p^{2} T^{4} - 11 p^{6} T^{5} + 19 p^{11} T^{6} + p^{15} T^{7} + p^{20} T^{8}$
	5	$C_2 \wr S_4$	$1 + 8 T + 4641 T^{2} + 522 T^{3} + 15201088 T^{4} + 522 p^{5} T^{5} + 4641 p^{10} T^{6} + 8 p^{15} T^{7} + p^{20} T^{8}$
	7	$C_2 \wr S_4$	$1 + 142 T + 35741 T^{2} + 6055818 T^{3} + 627818036 T^{4} + 6055818 p^{5} T^{5} + 35741 p^{10} T^{6} + 142 p^{15} T^{7} + p^{20} T^{8}$
	11	$C_2 \wr S_4$	$1 + 714 T + 448453 T^{2} + 212157030 T^{3} + 100726171004 T^{4} + 212157030 p^{5} T^{5} + 448453 p^{10} T^{6} + 714 p^{15} T^{7} + p^{20} T^{8}$
	13	$C_2 \wr S_4$	$1 - 74 T + 390128 T^{2} + 2718186 p T^{3} + 172797581198 T^{4} + 2718186 p^{6} T^{5} + 390128 p^{10} T^{6} - 74 p^{15} T^{7} + p^{20} T^{8}$
	17	$C_2 \wr S_4$	$1 + 3690 T + 10315729 T^{2} + 17980630638 T^{3} + 25584452309804 T^{4} + 17980630638 p^{5} T^{5} + 10315729 p^{10} T^{6} + 3690 p^{15} T^{7} + p^{20} T^{8}$
	23	$C_2 \wr S_4$	$1 - 862 T + 13814372 T^{2} - 19547784758 T^{3} + 113350258181846 T^{4} - 19547784758 p^{5} T^{5} + 13814372 p^{10} T^{6} - 862 p^{15} T^{7} + p^{20} T^{8}$
	29	$C_2 \wr S_4$	$1 + 992 T + 35667164 T^{2} + 1709314400 p T^{3} + 734484878068310 T^{4} + 1709314400 p^{6} T^{5} + 35667164 p^{10} T^{6} + 992 p^{15} T^{7} + p^{20} T^{8}$
	31	$C_2 \wr S_4$	$1 - 7382 T + 68912048 T^{2} - 398657804358 T^{3} + 2433438794556542 T^{4} - 398657804358 p^{5} T^{5} + 68912048 p^{10} T^{6} - 7382 p^{15} T^{7} + p^{20} T^{8}$
	37	$C_2 \wr S_4$	$1 + 19114 T + 9619124 p T^{2} + 3763460276310 T^{3} + 38837605627427894 T^{4} + 3763460276310 p^{5} T^{5} + 9619124 p^{11} T^{6} + 19114 p^{15} T^{7} + p^{20} T^{8}$
	41	$C_2 \wr S_4$	$1 - 5792 T + 184730688 T^{2} - 1152962318688 T^{3} + 35974773285629662 T^{4} - 1152962318688 p^{5} T^{5} + 184730688 p^{10} T^{6} - 5792 p^{15} T^{7} + p^{20} T^{8}$
	43	$C_2 \wr S_4$	$1 + 18634 T + 623524237 T^{2} + 7714523409682 T^{3} + 140468611843393948 T^{4} + 7714523409682 p^{5} T^{5} + 623524237 p^{10} T^{6} + 18634 p^{15} T^{7} + p^{20} T^{8}$
	47	$C_2 \wr S_4$	$1 + 35424 T + 1120098161 T^{2} + 21848880105198 T^{3} + 400407438721749108 T^{4} + 21848880105198 p^{5} T^{5} + 1120098161 p^{10} T^{6} + 35424 p^{15} T^{7} + p^{20} T^{8}$
	53	$C_2 \wr S_4$	$1 - 20256 T + 1403246860 T^{2} - 19678903472736 T^{3} + 818825687021028566 T^{4} - 19678903472736 p^{5} T^{5} + 1403246860 p^{10} T^{6} - 20256 p^{15} T^{7} + p^{20} T^{8}$
	59	$C_2 \wr S_4$	$1 - 76944 T + 4275207964 T^{2} - 165416309409936 T^{3} + 5007067871234044790 T^{4} - 165416309409936 p^{5} T^{5} + 4275207964 p^{10} T^{6} - 76944 p^{15} T^{7} + p^{20} T^{8}$
	61	$C_2 \wr S_4$	$1 - 11562 T + 2658185633 T^{2} - 22203715050858 T^{3} + 3131191588175453292 T^{4} - 22203715050858 p^{5} T^{5} + 2658185633 p^{10} T^{6} - 11562 p^{15} T^{7} + p^{20} T^{8}$
	67	$C_2 \wr S_4$	$1 - 110044 T + 8559033868 T^{2} - 454486933448572 T^{3} + 19453088667414801526 T^{4} - 454486933448572 p^{5} T^{5} + 8559033868 p^{10} T^{6} - 110044 p^{15} T^{7} + p^{20} T^{8}$
	71	$C_2 \wr S_4$	$1 - 71184 T + 2740541228 T^{2} - 50169050116560 T^{3} + 1927900005464350854 T^{4} - 50169050116560 p^{5} T^{5} + 2740541228 p^{10} T^{6} - 71184 p^{15} T^{7} + p^{20} T^{8}$
	73	$C_2 \wr S_4$	$1 + 138830 T + 14250815833 T^{2} + 933095031511802 T^{3} + 50238501223226001580 T^{4} + 933095031511802 p^{5} T^{5} + 14250815833 p^{10} T^{6} + 138830 p^{15} T^{7} + p^{20} T^{8}$
	79	$C_2 \wr S_4$	$1 - 59470 T + 2671253396 T^{2} + 110704118014410 T^{3} - 5424305011173132394 T^{4} + 110704118014410 p^{5} T^{5} + 2671253396 p^{10} T^{6} - 59470 p^{15} T^{7} + p^{20} T^{8}$
	83	$C_2 \wr S_4$	$1 + 191600 T + 26105339628 T^{2} + 2374053004024560 T^{3} +$ $17\!\cdots\!58$ $T^{4} + 2374053004024560 p^{5} T^{5} + 26105339628 p^{10} T^{6} + 191600 p^{15} T^{7} + p^{20} T^{8}$
	89	$C_2 \wr S_4$	$1 + 158920 T + 19242878460 T^{2} + 1772160038528184 T^{3} +$ $15\!\cdots\!58$ $T^{4} + 1772160038528184 p^{5} T^{5} + 19242878460 p^{10} T^{6} + 158920 p^{15} T^{7} + p^{20} T^{8}$
	97	$C_2 \wr S_4$	$1 + 47460 T + 30660377492 T^{2} + 1213862934386844 T^{3} +$ $37\!\cdots\!30$ $T^{4} + 1213862934386844 p^{5} T^{5} + 30660377492 p^{10} T^{6} + 47460 p^{15} T^{7} + p^{20} T^{8}$
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L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.90469963380845185744929662125, −6.73655453604572571852581987658, −6.57853585080754224334604869196, −6.07607774038294143172760149429, −5.96700091963856050016608607066, −5.54314279787561233059408617971, −5.21774333059564280829029020809, −5.00369585269591043171477730822, −4.99369125129293065654370007501, −4.51844401243956478034409314959, −4.44902178681620811872317105957, −4.27880778806936406504839403627, −3.84382623614995440227751968004, −3.73958523140937303457389658007, −3.54724362005096921113186812931, −3.27536878444024553938185776667, −3.00145777916894975989453214451, −2.52257604945653301922592626349, −2.51327375339796087185632981849, −2.35360073910634656365312358778, −2.06731655320669800613842470062, −1.80080843472178054560564813602, −1.49660034736858602648140128618, −1.07741445426476957570259051858, −0.73240527763168022963818017047, 0, 0, 0, 0, 0.73240527763168022963818017047, 1.07741445426476957570259051858, 1.49660034736858602648140128618, 1.80080843472178054560564813602, 2.06731655320669800613842470062, 2.35360073910634656365312358778, 2.51327375339796087185632981849, 2.52257604945653301922592626349, 3.00145777916894975989453214451, 3.27536878444024553938185776667, 3.54724362005096921113186812931, 3.73958523140937303457389658007, 3.84382623614995440227751968004, 4.27880778806936406504839403627, 4.44902178681620811872317105957, 4.51844401243956478034409314959, 4.99369125129293065654370007501, 5.00369585269591043171477730822, 5.21774333059564280829029020809, 5.54314279787561233059408617971, 5.96700091963856050016608607066, 6.07607774038294143172760149429, 6.57853585080754224334604869196, 6.73655453604572571852581987658, 6.90469963380845185744929662125

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more

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	8-1083e4-1.1-c5e4-0-0
Degree	$8$
Conductor	$1.376\times 10^{12}$
Sign	$1$
Analytic cond.	$9.10240\times 10^{8}$
Root an. cond.	$13.1793$
Motivic weight	$5$
Arithmetic	yes
Rational	yes
Primitive	no
Self-dual	yes
Analytic rank	$4$

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