LMFDB - L-function 16-147e8-1.1-c5e8-0-0

Dirichlet series

L(s) = 1

− 3·2^-s + 36·3^-s + 34·4^-s − 108·6^-s − 75·8^-s + 486·9^-s − 402·11^-s + 1.22e3·12^-s − 924·13^-s + 251·16^-s + 276·17^-s − 1.45e3·18^-s + 510·19^-s + 1.20e3·22^-s − 6.90e3·23^-s − 2.70e3·24^-s + 4.84e3·25^-s + 2.77e3·26^-s + 1.08e3·29^-s − 6.41e3·31^-s − 2.34e3·32^-s − 1.44e4·33^-s − 828·34^-s + 1.65e4·36^-s − 1.52e4·37^-s − 1.53e3·38^-s − 3.32e4·39^-s + ⋯

L(s) = 1

− 0.530·2^-s + 2.30·3^-s + 1.06·4^-s − 1.22·6^-s − 0.414·8^-s + 2·9^-s − 1.00·11^-s + 2.45·12^-s − 1.51·13^-s + 0.245·16^-s + 0.231·17^-s − 1.06·18^-s + 0.324·19^-s + 0.531·22^-s − 2.71·23^-s − 0.956·24^-s + 1.54·25^-s + 0.804·26^-s + 0.238·29^-s − 1.19·31^-s − 0.404·32^-s − 2.31·33^-s − 0.122·34^-s + 17/8·36^-s − 1.83·37^-s − 0.171·38^-s − 3.50·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$3^{8} \cdot 7^{16}$
Sign:	$1$
Analytic conductor:	$9.54603\times 10^{10}$
Root analytic conductor:	$4.85555$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 3^{8} \cdot 7^{16} ,\ ( \ : [5/2]^{8} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$31.66179147$
$L(\frac12)$	$\approx$	$31.66179147$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ ( 1 - p^{2} T + p^{4} T^{2} )^{4} $
	7	$ 1 $
good	2	$ 1 + 3 T - 25 T^{2} - 51 p T^{3} + 259 p T^{4} + 1185 p^{2} T^{5} + 14739 p^{2} T^{6} + 1833 p^{3} T^{7} - 116575 p^{4} T^{8} + 1833 p^{8} T^{9} + 14739 p^{12} T^{10} + 1185 p^{17} T^{11} + 259 p^{21} T^{12} - 51 p^{26} T^{13} - 25 p^{30} T^{14} + 3 p^{35} T^{15} + p^{40} T^{16} $
	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

−5.05307791406769457207849168895, −4.92712954383861458374969040833, −4.72957975982543698217639054564, −4.31752645131620556006845878574, −4.30261160042549693159129301285, −3.91480400933301333557298442757, −3.79577306955930683834142523529, −3.61824299976399470164633089295, −3.48683772359282345589540075745, −3.42385713808556408189656861097, −3.23844859942235457790333170546, −2.88507736115406102639415293585, −2.81465718732409702506243497772, −2.41655969477833661649222774965, −2.36397423999022205422348389040, −2.17325847057059899031331605466, −2.15694558860676772556769789913, −1.94483381893942027133782249754, −1.93027652560405659351694265794, −1.46994270815991637902921007129, −1.18639935162307989511689772319, −0.62445258124539311716451783425, −0.55243391021753936573227649399, −0.54729771755197061543776263797, −0.38892228681373068113390755483, 0.38892228681373068113390755483, 0.54729771755197061543776263797, 0.55243391021753936573227649399, 0.62445258124539311716451783425, 1.18639935162307989511689772319, 1.46994270815991637902921007129, 1.93027652560405659351694265794, 1.94483381893942027133782249754, 2.15694558860676772556769789913, 2.17325847057059899031331605466, 2.36397423999022205422348389040, 2.41655969477833661649222774965, 2.81465718732409702506243497772, 2.88507736115406102639415293585, 3.23844859942235457790333170546, 3.42385713808556408189656861097, 3.48683772359282345589540075745, 3.61824299976399470164633089295, 3.79577306955930683834142523529, 3.91480400933301333557298442757, 4.30261160042549693159129301285, 4.31752645131620556006845878574, 4.72957975982543698217639054564, 4.92712954383861458374969040833, 5.05307791406769457207849168895

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(31.66179147\)
\(L(\frac12)\)	\(\approx\)	\(31.66179147\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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