LMFDB - L-function 16-336e8-1.1-c5e8-0-2

Dirichlet series

L(s) = 1

+ 36·3^-s − 258·7^-s + 486·9^-s + 402·11^-s + 924·13^-s − 276·17^-s + 510·19^-s − 9.28e3·21^-s + 6.90e3·23^-s + 4.84e3·25^-s + 1.08e3·29^-s − 6.41e3·31^-s + 1.44e4·33^-s − 1.52e4·37^-s + 3.32e4·39^-s + 8.61e3·41^-s − 5.83e4·43^-s − 1.50e4·47^-s + 1.15e3·49^-s − 9.93e3·51^-s − 1.36e4·53^-s + 1.83e4·57^-s + 3.48e4·59^-s + 5.36e3·61^-s − 1.25e5·63^-s − 5.99e3·67^-s + 2.48e5·69^-s + ⋯

L(s) = 1

+ 2.30·3^-s − 1.99·7^-s + 2·9^-s + 1.00·11^-s + 1.51·13^-s − 0.231·17^-s + 0.324·19^-s − 4.59·21^-s + 2.71·23^-s + 1.54·25^-s + 0.238·29^-s − 1.19·31^-s + 2.31·33^-s − 1.83·37^-s + 3.50·39^-s + 0.800·41^-s − 4.81·43^-s − 0.994·47^-s + 0.0687·49^-s − 0.534·51^-s − 0.669·53^-s + 0.748·57^-s + 1.30·59^-s + 0.184·61^-s − 3.98·63^-s − 0.163·67^-s + 6.28·69^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{32} \cdot 3^{8} \cdot 7^{8}$
Sign:	$1$
Analytic conductor:	$7.11211\times 10^{13}$
Root analytic conductor:	$7.34091$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{32} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$6.440714306$
$L(\frac12)$	$\approx$	$6.440714306$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ ( 1 - p^{2} T + p^{4} T^{2} )^{4} $
	7	$ 1 + 258 T + 9344 p T^{2} + 251256 p^{2} T^{3} + 4790655 p^{3} T^{4} + 251256 p^{7} T^{5} + 9344 p^{11} T^{6} + 258 p^{15} T^{7} + p^{20} T^{8} $
good	5	$ 1 - 4843 T^{2} + 24216 p^{2} T^{3} + 15905393 T^{4} - 96476544 p^{2} T^{5} + 149656086942 T^{6} + 275087633352 p^{2} T^{7} - 553760753377714 T^{8} + 275087633352 p^{7} T^{9} + 149656086942 p^{10} T^{10} - 96476544 p^{17} T^{11} + 15905393 p^{20} T^{12} + 24216 p^{27} T^{13} - 4843 p^{30} T^{14} + p^{40} T^{16} $
	11	$ 1 - 402 T - 244237 T^{2} + 23685786 T^{3} + 49988093357 T^{4} + 825774680136 p T^{5} - 7585190415099126 T^{6} - 907765850007682500 T^{7} + $$95\!\cdots\!06$$ T^{8} - 907765850007682500 p^{5} T^{9} - 7585190415099126 p^{10} T^{10} + 825774680136 p^{16} T^{11} + 49988093357 p^{20} T^{12} + 23685786 p^{25} T^{13} - 244237 p^{30} T^{14} - 402 p^{35} T^{15} + p^{40} T^{16} $
	13	$ ( 1 - 462 T + 336749 T^{2} + 500754 T^{3} + 123849672672 T^{4} + 500754 p^{5} T^{5} + 336749 p^{10} T^{6} - 462 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	17	$ 1 + 276 T - 4008500 T^{2} + 565843800 T^{3} + 9089867656090 T^{4} - 2604033925498692 T^{5} - 14267240611484004272 T^{6} + $$19\!\cdots\!84$$ T^{7} + $$19\!\cdots\!07$$ T^{8} + $$19\!\cdots\!84$$ p^{5} T^{9} - 14267240611484004272 p^{10} T^{10} - 2604033925498692 p^{15} T^{11} + 9089867656090 p^{20} T^{12} + 565843800 p^{25} T^{13} - 4008500 p^{30} T^{14} + 276 p^{35} T^{15} + p^{40} T^{16} $
	19	$ 1 - 510 T - 3747425 T^{2} + 3937377630 T^{3} - 440016487199 T^{4} - 4115411338726080 T^{5} - 5704973911539330050 T^{6} - $$57\!\cdots\!00$$ T^{7} + $$73\!\cdots\!50$$ T^{8} - $$57\!\cdots\!00$$ p^{5} T^{9} - 5704973911539330050 p^{10} T^{10} - 4115411338726080 p^{15} T^{11} - 440016487199 p^{20} T^{12} + 3937377630 p^{25} T^{13} - 3747425 p^{30} T^{14} - 510 p^{35} T^{15} + p^{40} T^{16} $
	23	$ 1 - 300 p T + 14742772 T^{2} + 5434854360 T^{3} - 64144356593702 T^{4} + 173260413144408420 T^{5} - $$23\!\cdots\!64$$ T^{6} - $$19\!\cdots\!40$$ T^{7} + $$99\!\cdots\!79$$ T^{8} - $$19\!\cdots\!40$$ p^{5} T^{9} - $$23\!\cdots\!64$$ p^{10} T^{10} + 173260413144408420 p^{15} T^{11} - 64144356593702 p^{20} T^{12} + 5434854360 p^{25} T^{13} + 14742772 p^{30} T^{14} - 300 p^{36} T^{15} + p^{40} T^{16} $
	29	$ ( 1 - 540 T + 29394199 T^{2} + 29299685892 T^{3} + 772430149366772 T^{4} + 29299685892 p^{5} T^{5} + 29394199 p^{10} T^{6} - 540 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	31	$ 1 + 6410 T - 55714636 T^{2} - 363334551632 T^{3} + 2227342403022119 T^{4} + 10547841526395999412 T^{5} - $$75\!\cdots\!84$$ T^{6} - $$10\!\cdots\!10$$ T^{7} + $$25\!\cdots\!76$$ T^{8} - $$10\!\cdots\!10$$ p^{5} T^{9} - $$75\!\cdots\!84$$ p^{10} T^{10} + 10547841526395999412 p^{15} T^{11} + 2227342403022119 p^{20} T^{12} - 363334551632 p^{25} T^{13} - 55714636 p^{30} T^{14} + 6410 p^{35} T^{15} + p^{40} T^{16} $
	37	$ 1 + 15250 T + 2042879 T^{2} + 164587155110 T^{3} + 277810073119073 p T^{4} - 25563809512056184540 T^{5} - $$73\!\cdots\!82$$ T^{6} - $$25\!\cdots\!80$$ T^{7} - $$18\!\cdots\!10$$ T^{8} - $$25\!\cdots\!80$$ p^{5} T^{9} - $$73\!\cdots\!82$$ p^{10} T^{10} - 25563809512056184540 p^{15} T^{11} + 277810073119073 p^{21} T^{12} + 164587155110 p^{25} T^{13} + 2042879 p^{30} T^{14} + 15250 p^{35} T^{15} + p^{40} T^{16} $
	41	$ ( 1 - 4308 T + 270683560 T^{2} - 2598901038204 T^{3} + 34018560333944366 T^{4} - 2598901038204 p^{5} T^{5} + 270683560 p^{10} T^{6} - 4308 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	43	$ ( 1 + 29198 T + 787995265 T^{2} + 13076978932730 T^{3} + 187469286625894360 T^{4} + 13076978932730 p^{5} T^{5} + 787995265 p^{10} T^{6} + 29198 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	47	$ 1 + 15060 T - 741226172 T^{2} - 6055210491384 T^{3} + 443152946335783210 T^{4} + $$21\!\cdots\!28$$ T^{5} - $$15\!\cdots\!68$$ T^{6} - $$13\!\cdots\!48$$ T^{7} + $$43\!\cdots\!71$$ T^{8} - $$13\!\cdots\!48$$ p^{5} T^{9} - $$15\!\cdots\!68$$ p^{10} T^{10} + $$21\!\cdots\!28$$ p^{15} T^{11} + 443152946335783210 p^{20} T^{12} - 6055210491384 p^{25} T^{13} - 741226172 p^{30} T^{14} + 15060 p^{35} T^{15} + p^{40} T^{16} $
	53	$ 1 + 13692 T - 987403679 T^{2} - 2191455111540 T^{3} + 630826063270172605 T^{4} - $$25\!\cdots\!24$$ T^{5} - $$29\!\cdots\!74$$ T^{6} + $$54\!\cdots\!28$$ T^{7} + $$11\!\cdots\!06$$ T^{8} + $$54\!\cdots\!28$$ p^{5} T^{9} - $$29\!\cdots\!74$$ p^{10} T^{10} - $$25\!\cdots\!24$$ p^{15} T^{11} + 630826063270172605 p^{20} T^{12} - 2191455111540 p^{25} T^{13} - 987403679 p^{30} T^{14} + 13692 p^{35} T^{15} + p^{40} T^{16} $
	59	$ 1 - 34830 T - 1834824737 T^{2} + 42321414019590 T^{3} + 3370913750880697081 T^{4} - $$47\!\cdots\!20$$ T^{5} - $$33\!\cdots\!82$$ T^{6} + $$90\!\cdots\!20$$ T^{7} + $$30\!\cdots\!74$$ T^{8} + $$90\!\cdots\!20$$ p^{5} T^{9} - $$33\!\cdots\!82$$ p^{10} T^{10} - $$47\!\cdots\!20$$ p^{15} T^{11} + 3370913750880697081 p^{20} T^{12} + 42321414019590 p^{25} T^{13} - 1834824737 p^{30} T^{14} - 34830 p^{35} T^{15} + p^{40} T^{16} $
	61	$ 1 - 5364 T - 2817016532 T^{2} + 5158230942312 T^{3} + 4645398449535186778 T^{4} - $$12\!\cdots\!24$$ T^{5} - $$55\!\cdots\!52$$ T^{6} + $$26\!\cdots\!00$$ T^{7} + $$52\!\cdots\!87$$ T^{8} + $$26\!\cdots\!00$$ p^{5} T^{9} - $$55\!\cdots\!52$$ p^{10} T^{10} - $$12\!\cdots\!24$$ p^{15} T^{11} + 4645398449535186778 p^{20} T^{12} + 5158230942312 p^{25} T^{13} - 2817016532 p^{30} T^{14} - 5364 p^{35} T^{15} + p^{40} T^{16} $
	67	$ 1 + 5994 T - 2518577501 T^{2} - 43127704637370 T^{3} + 2548139821971114109 T^{4} + $$68\!\cdots\!04$$ T^{5} + $$21\!\cdots\!62$$ T^{6} - $$47\!\cdots\!72$$ T^{7} - $$17\!\cdots\!02$$ T^{8} - $$47\!\cdots\!72$$ p^{5} T^{9} + $$21\!\cdots\!62$$ p^{10} T^{10} + $$68\!\cdots\!04$$ p^{15} T^{11} + 2548139821971114109 p^{20} T^{12} - 43127704637370 p^{25} T^{13} - 2518577501 p^{30} T^{14} + 5994 p^{35} T^{15} + p^{40} T^{16} $
	71	$ ( 1 + 89268 T + 8738662172 T^{2} + 466802240277492 T^{3} + 25044546792022133910 T^{4} + 466802240277492 p^{5} T^{5} + 8738662172 p^{10} T^{6} + 89268 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	73	$ 1 + 59638 T - 5098535509 T^{2} - 184967914738150 T^{3} + 26836978407770089433 T^{4} + $$52\!\cdots\!12$$ T^{5} - $$81\!\cdots\!50$$ T^{6} - $$30\!\cdots\!84$$ T^{7} + $$20\!\cdots\!42$$ T^{8} - $$30\!\cdots\!84$$ p^{5} T^{9} - $$81\!\cdots\!50$$ p^{10} T^{10} + $$52\!\cdots\!12$$ p^{15} T^{11} + 26836978407770089433 p^{20} T^{12} - 184967914738150 p^{25} T^{13} - 5098535509 p^{30} T^{14} + 59638 p^{35} T^{15} + p^{40} T^{16} $
	79	$ 1 + 44062 T - 6584960844 T^{2} - 467510535225040 T^{3} + 20422039882069416887 T^{4} + $$19\!\cdots\!32$$ T^{5} - $$53\!\cdots\!72$$ T^{6} - $$30\!\cdots\!70$$ T^{7} - $$76\!\cdots\!68$$ T^{8} - $$30\!\cdots\!70$$ p^{5} T^{9} - $$53\!\cdots\!72$$ p^{10} T^{10} + $$19\!\cdots\!32$$ p^{15} T^{11} + 20422039882069416887 p^{20} T^{12} - 467510535225040 p^{25} T^{13} - 6584960844 p^{30} T^{14} + 44062 p^{35} T^{15} + p^{40} T^{16} $
	83	$ ( 1 - 208446 T + 23363412401 T^{2} - 1724627286611514 T^{3} + $$11\!\cdots\!56$$ T^{4} - 1724627286611514 p^{5} T^{5} + 23363412401 p^{10} T^{6} - 208446 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	89	$ 1 - 77520 T + 479326112 T^{2} - 1563841943328288 T^{3} + $$12\!\cdots\!70$$ T^{4} - $$11\!\cdots\!72$$ T^{5} + $$11\!\cdots\!40$$ T^{6} - $$89\!\cdots\!20$$ T^{7} + $$66\!\cdots\!67$$ T^{8} - $$89\!\cdots\!20$$ p^{5} T^{9} + $$11\!\cdots\!40$$ p^{10} T^{10} - $$11\!\cdots\!72$$ p^{15} T^{11} + $$12\!\cdots\!70$$ p^{20} T^{12} - 1563841943328288 p^{25} T^{13} + 479326112 p^{30} T^{14} - 77520 p^{35} T^{15} + p^{40} T^{16} $
	97	$ ( 1 + 188630 T + 43620869129 T^{2} + 4760435860011510 T^{3} + $$59\!\cdots\!64$$ T^{4} + 4760435860011510 p^{5} T^{5} + 43620869129 p^{10} T^{6} + 188630 p^{15} T^{7} + p^{20} T^{8} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.92784768087516316022765221630, −3.83235626691317496463558393072, −3.79375999447260143362188845650, −3.77621961891688628790151425506, −3.62753007918037661067905028404, −3.32059230913794734718878888812, −3.18652749002496854757248991718, −3.16981203884020678710840393660, −3.01690650015013815391708820836, −2.96865418977090041309928911677, −2.74058972691527770355361610593, −2.72614166914596173409571298870, −2.56178306353535897898732439532, −2.07499459641304160013335754294, −1.89160443187907557869238342379, −1.85303699278435654611264686534, −1.57395270163067644911969534405, −1.54572592599928544773981665960, −1.40741748889352541167801562551, −1.18554491418263021655227918587, −0.933731316837381460171144728311, −0.73713304501588895031437813525, −0.40085532481543234424406578077, −0.34035693491137297772179044406, −0.12705840911659962893721265145, 0.12705840911659962893721265145, 0.34035693491137297772179044406, 0.40085532481543234424406578077, 0.73713304501588895031437813525, 0.933731316837381460171144728311, 1.18554491418263021655227918587, 1.40741748889352541167801562551, 1.54572592599928544773981665960, 1.57395270163067644911969534405, 1.85303699278435654611264686534, 1.89160443187907557869238342379, 2.07499459641304160013335754294, 2.56178306353535897898732439532, 2.72614166914596173409571298870, 2.74058972691527770355361610593, 2.96865418977090041309928911677, 3.01690650015013815391708820836, 3.16981203884020678710840393660, 3.18652749002496854757248991718, 3.32059230913794734718878888812, 3.62753007918037661067905028404, 3.77621961891688628790151425506, 3.79375999447260143362188845650, 3.83235626691317496463558393072, 3.92784768087516316022765221630

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

Plot not available for L-functions of degree greater than 10.

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(6.440714306\)
\(L(\frac12)\)	\(\approx\)	\(6.440714306\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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