LMFDB - L-function 16-588e8-1.1-c5e8-0-1

Dirichlet series

L(s) = 1

− 36·3^-s + 486·9^-s − 462·11^-s + 1.20e3·13^-s − 228·17^-s − 358·19^-s − 2.14e3·23^-s + 3.52e3·25^-s − 1.10e4·29^-s − 830·31^-s + 1.66e4·33^-s − 3.91e3·37^-s − 4.33e4·39^-s + 1.66e4·41^-s − 2.90e4·43^-s − 4.17e4·47^-s + 8.20e3·51^-s + 2.21e4·53^-s + 1.28e4·57^-s − 3.28e4·59^-s − 8.37e4·61^-s − 8.00e4·67^-s + 7.73e4·69^-s + 8.95e4·71^-s + 2.24e4·73^-s − 1.26e5·75^-s − 7.52e4·79^-s + ⋯

L(s) = 1

− 2.30·3^-s + 2·9^-s − 1.15·11^-s + 1.97·13^-s − 0.191·17^-s − 0.227·19^-s − 0.846·23^-s + 1.12·25^-s − 2.44·29^-s − 0.155·31^-s + 2.65·33^-s − 0.470·37^-s − 4.56·39^-s + 1.54·41^-s − 2.39·43^-s − 2.75·47^-s + 0.441·51^-s + 1.08·53^-s + 0.525·57^-s − 1.22·59^-s − 2.88·61^-s − 2.17·67^-s + 1.95·69^-s + 2.10·71^-s + 0.493·73^-s − 2.60·75^-s − 1.35·79^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$16$
Conductor:	$2^{16} \cdot 3^{8} \cdot 7^{16}$
Sign:	$1$
Analytic conductor:	$6.25609\times 10^{15}$
Root analytic conductor:	$9.71111$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{16} \cdot 3^{8} \cdot 7^{16} ,\ ( \ : [5/2]^{8} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$2.254716139$
$L(\frac12)$	$\approx$	$2.254716139$
$L(\frac{7}{2})$		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 $
good	5	$ 1 - 3523 T^{2} - 264 p^{4} T^{3} - 839647 T^{4} + 877536 p^{4} T^{5} + 28930950702 T^{6} - 1157319768 p^{4} T^{7} - 62249635398274 T^{8} - 1157319768 p^{9} T^{9} + 28930950702 p^{10} T^{10} + 877536 p^{19} T^{11} - 839647 p^{20} T^{12} - 264 p^{29} T^{13} - 3523 p^{30} T^{14} + p^{40} T^{16} $
	11	$ 1 + 42 p T - 55261 T^{2} + 23882490 T^{3} + 30394966829 T^{4} + 4313182607736 T^{5} + 7711402699001610 T^{6} + 1745859627280956732 T^{7} - $$81\!\cdots\!62$$ T^{8} + 1745859627280956732 p^{5} T^{9} + 7711402699001610 p^{10} T^{10} + 4313182607736 p^{15} T^{11} + 30394966829 p^{20} T^{12} + 23882490 p^{25} T^{13} - 55261 p^{30} T^{14} + 42 p^{36} T^{15} + p^{40} T^{16} $
	13	$ ( 1 - 602 T - 12679 p T^{2} - 145984794 T^{3} + 357680241104 T^{4} - 145984794 p^{5} T^{5} - 12679 p^{11} T^{6} - 602 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	17	$ 1 + 228 T - 3817300 T^{2} - 1812530376 T^{3} + 7723728660698 T^{4} + 3961559277827628 T^{5} - 9874383121499354160 T^{6} - $$30\!\cdots\!88$$ T^{7} + $$11\!\cdots\!63$$ T^{8} - $$30\!\cdots\!88$$ p^{5} T^{9} - 9874383121499354160 p^{10} T^{10} + 3961559277827628 p^{15} T^{11} + 7723728660698 p^{20} T^{12} - 1812530376 p^{25} T^{13} - 3817300 p^{30} T^{14} + 228 p^{35} T^{15} + p^{40} T^{16} $
	19	$ 1 + 358 T - 7237113 T^{2} - 2935172086 T^{3} + 28171845756353 T^{4} + 9461877710293488 T^{5} - 89100991373462587298 T^{6} - $$10\!\cdots\!20$$ T^{7} + $$24\!\cdots\!42$$ T^{8} - $$10\!\cdots\!20$$ p^{5} T^{9} - 89100991373462587298 p^{10} T^{10} + 9461877710293488 p^{15} T^{11} + 28171845756353 p^{20} T^{12} - 2935172086 p^{25} T^{13} - 7237113 p^{30} T^{14} + 358 p^{35} T^{15} + p^{40} T^{16} $
	23	$ 1 + 2148 T - 12238924 T^{2} + 4077683208 T^{3} + 128948594947802 T^{4} - 143919044174116500 T^{5} - 26075111207770425456 p T^{6} + $$52\!\cdots\!72$$ T^{7} + $$18\!\cdots\!31$$ T^{8} + $$52\!\cdots\!72$$ p^{5} T^{9} - 26075111207770425456 p^{11} T^{10} - 143919044174116500 p^{15} T^{11} + 128948594947802 p^{20} T^{12} + 4077683208 p^{25} T^{13} - 12238924 p^{30} T^{14} + 2148 p^{35} T^{15} + p^{40} T^{16} $
	29	$ ( 1 + 5532 T + 56710687 T^{2} + 131401968540 T^{3} + 1151610033241988 T^{4} + 131401968540 p^{5} T^{5} + 56710687 p^{10} T^{6} + 5532 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	31	$ 1 + 830 T - 30566692 T^{2} + 460416975904 T^{3} + 906611510861135 T^{4} - 13425215480714328836 T^{5} + $$95\!\cdots\!40$$ T^{6} + $$37\!\cdots\!30$$ T^{7} - $$27\!\cdots\!28$$ T^{8} + $$37\!\cdots\!30$$ p^{5} T^{9} + $$95\!\cdots\!40$$ p^{10} T^{10} - 13425215480714328836 p^{15} T^{11} + 906611510861135 p^{20} T^{12} + 460416975904 p^{25} T^{13} - 30566692 p^{30} T^{14} + 830 p^{35} T^{15} + p^{40} T^{16} $
	37	$ 1 + 3914 T - 1798245 p T^{2} - 1097998070882 T^{3} - 1825206898694683 T^{4} + 62220290884192688868 T^{5} + $$62\!\cdots\!66$$ T^{6} - $$11\!\cdots\!88$$ T^{7} - $$46\!\cdots\!38$$ T^{8} - $$11\!\cdots\!88$$ p^{5} T^{9} + $$62\!\cdots\!66$$ p^{10} T^{10} + 62220290884192688868 p^{15} T^{11} - 1825206898694683 p^{20} T^{12} - 1097998070882 p^{25} T^{13} - 1798245 p^{31} T^{14} + 3914 p^{35} T^{15} + p^{40} T^{16} $
	41	$ ( 1 - 8316 T + 131719112 T^{2} + 1793338464780 T^{3} - 11507939827736466 T^{4} + 1793338464780 p^{5} T^{5} + 131719112 p^{10} T^{6} - 8316 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	43	$ ( 1 + 14518 T + 553805833 T^{2} + 5594375858722 T^{3} + 119245786138973608 T^{4} + 5594375858722 p^{5} T^{5} + 553805833 p^{10} T^{6} + 14518 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	47	$ 1 + 41700 T + 537485524 T^{2} + 6358172548200 T^{3} + 162169996689714634 T^{4} + $$49\!\cdots\!00$$ T^{5} - $$48\!\cdots\!44$$ T^{6} - $$17\!\cdots\!00$$ p T^{7} - $$10\!\cdots\!61$$ T^{8} - $$17\!\cdots\!00$$ p^{6} T^{9} - $$48\!\cdots\!44$$ p^{10} T^{10} + $$49\!\cdots\!00$$ p^{15} T^{11} + 162169996689714634 p^{20} T^{12} + 6358172548200 p^{25} T^{13} + 537485524 p^{30} T^{14} + 41700 p^{35} T^{15} + p^{40} T^{16} $
	53	$ 1 - 22164 T - 121257815 T^{2} - 31884451303188 T^{3} + 713358687755937853 T^{4} + $$34\!\cdots\!36$$ T^{5} + $$46\!\cdots\!10$$ T^{6} - $$99\!\cdots\!24$$ T^{7} - $$41\!\cdots\!02$$ T^{8} - $$99\!\cdots\!24$$ p^{5} T^{9} + $$46\!\cdots\!10$$ p^{10} T^{10} + $$34\!\cdots\!36$$ p^{15} T^{11} + 713358687755937853 p^{20} T^{12} - 31884451303188 p^{25} T^{13} - 121257815 p^{30} T^{14} - 22164 p^{35} T^{15} + p^{40} T^{16} $
	59	$ 1 + 32886 T - 1083783697 T^{2} - 35352420196734 T^{3} + 734569909452509033 T^{4} + $$86\!\cdots\!92$$ T^{5} - $$96\!\cdots\!02$$ T^{6} + $$76\!\cdots\!28$$ T^{7} + $$95\!\cdots\!14$$ T^{8} + $$76\!\cdots\!28$$ p^{5} T^{9} - $$96\!\cdots\!02$$ p^{10} T^{10} + $$86\!\cdots\!92$$ p^{15} T^{11} + 734569909452509033 p^{20} T^{12} - 35352420196734 p^{25} T^{13} - 1083783697 p^{30} T^{14} + 32886 p^{35} T^{15} + p^{40} T^{16} $
	61	$ 1 + 83732 T + 1872200780 T^{2} + 2539765176280 T^{3} + 1308252811325909786 T^{4} + $$75\!\cdots\!52$$ T^{5} + $$99\!\cdots\!48$$ T^{6} + $$38\!\cdots\!36$$ T^{7} + $$21\!\cdots\!19$$ T^{8} + $$38\!\cdots\!36$$ p^{5} T^{9} + $$99\!\cdots\!48$$ p^{10} T^{10} + $$75\!\cdots\!52$$ p^{15} T^{11} + 1308252811325909786 p^{20} T^{12} + 2539765176280 p^{25} T^{13} + 1872200780 p^{30} T^{14} + 83732 p^{35} T^{15} + p^{40} T^{16} $
	67	$ 1 + 80034 T + 295144795 T^{2} - 67594226392002 T^{3} + 2776430208831340477 T^{4} + $$10\!\cdots\!48$$ T^{5} - $$54\!\cdots\!94$$ T^{6} - $$84\!\cdots\!88$$ T^{7} + $$64\!\cdots\!82$$ T^{8} - $$84\!\cdots\!88$$ p^{5} T^{9} - $$54\!\cdots\!94$$ p^{10} T^{10} + $$10\!\cdots\!48$$ p^{15} T^{11} + 2776430208831340477 p^{20} T^{12} - 67594226392002 p^{25} T^{13} + 295144795 p^{30} T^{14} + 80034 p^{35} T^{15} + p^{40} T^{16} $
	71	$ ( 1 - 44772 T + 5603702956 T^{2} - 219545846683812 T^{3} + 14130153190557486902 T^{4} - 219545846683812 p^{5} T^{5} + 5603702956 p^{10} T^{6} - 44772 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	73	$ 1 - 22470 T - 3227884709 T^{2} - 55184471402730 T^{3} + 5281219798633591705 T^{4} + $$29\!\cdots\!40$$ T^{5} + $$52\!\cdots\!98$$ T^{6} - $$45\!\cdots\!60$$ T^{7} - $$39\!\cdots\!94$$ T^{8} - $$45\!\cdots\!60$$ p^{5} T^{9} + $$52\!\cdots\!98$$ p^{10} T^{10} + $$29\!\cdots\!40$$ p^{15} T^{11} + 5281219798633591705 p^{20} T^{12} - 55184471402730 p^{25} T^{13} - 3227884709 p^{30} T^{14} - 22470 p^{35} T^{15} + p^{40} T^{16} $
	79	$ 1 + 75286 T - 4589904580 T^{2} - 271785361643200 T^{3} + 23588797479279670751 T^{4} + $$58\!\cdots\!56$$ T^{5} - $$88\!\cdots\!28$$ T^{6} - $$13\!\cdots\!78$$ T^{7} + $$19\!\cdots\!84$$ T^{8} - $$13\!\cdots\!78$$ p^{5} T^{9} - $$88\!\cdots\!28$$ p^{10} T^{10} + $$58\!\cdots\!56$$ p^{15} T^{11} + 23588797479279670751 p^{20} T^{12} - 271785361643200 p^{25} T^{13} - 4589904580 p^{30} T^{14} + 75286 p^{35} T^{15} + p^{40} T^{16} $
	83	$ ( 1 - 17418 T + 11662792993 T^{2} - 252243184391142 T^{3} + 61381737302005449608 T^{4} - 252243184391142 p^{5} T^{5} + 11662792993 p^{10} T^{6} - 17418 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	89	$ 1 + 28944 T - 4147498688 T^{2} - 127105851999456 T^{3} + 27498803712269304034 T^{4} + $$18\!\cdots\!24$$ T^{5} + $$34\!\cdots\!76$$ T^{6} + $$10\!\cdots\!68$$ T^{7} - $$15\!\cdots\!29$$ T^{8} + $$10\!\cdots\!68$$ p^{5} T^{9} + $$34\!\cdots\!76$$ p^{10} T^{10} + $$18\!\cdots\!24$$ p^{15} T^{11} + 27498803712269304034 p^{20} T^{12} - 127105851999456 p^{25} T^{13} - 4147498688 p^{30} T^{14} + 28944 p^{35} T^{15} + p^{40} T^{16} $
	97	$ ( 1 - 216678 T + 43602999305 T^{2} - 4974916579453302 T^{3} + $$56\!\cdots\!00$$ T^{4} - 4974916579453302 p^{5} T^{5} + 43602999305 p^{10} T^{6} - 216678 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

−3.78725611736600871477000197084, −3.40804613057975487160854921369, −3.39228616223431254588499237447, −3.38470857199495160400649404749, −3.38358124510480296654621372773, −3.22921224745443967011301324273, −3.11286854645197789354337787934, −3.09511539833759666465238796610, −2.68181713973450480556465538386, −2.30236535280910748664620517995, −2.27216269510107732894365325685, −2.25282808466384982199113150622, −1.93353275365786013877437808135, −1.78406034225996911245922485760, −1.75796981814511933371357216513, −1.66493122688237954049324871011, −1.46324361554790387047470707127, −1.26700399564779224186915229426, −0.937635321912667336934340148633, −0.76584839327575808548131041259, −0.70568049398851539419256735121, −0.51533397506853286279368896396, −0.44926658143144563230536709623, −0.26767343996333052369294636123, −0.16350537271178732751894324320, 0.16350537271178732751894324320, 0.26767343996333052369294636123, 0.44926658143144563230536709623, 0.51533397506853286279368896396, 0.70568049398851539419256735121, 0.76584839327575808548131041259, 0.937635321912667336934340148633, 1.26700399564779224186915229426, 1.46324361554790387047470707127, 1.66493122688237954049324871011, 1.75796981814511933371357216513, 1.78406034225996911245922485760, 1.93353275365786013877437808135, 2.25282808466384982199113150622, 2.27216269510107732894365325685, 2.30236535280910748664620517995, 2.68181713973450480556465538386, 3.09511539833759666465238796610, 3.11286854645197789354337787934, 3.22921224745443967011301324273, 3.38358124510480296654621372773, 3.38470857199495160400649404749, 3.39228616223431254588499237447, 3.40804613057975487160854921369, 3.78725611736600871477000197084

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(2.254716139\)
\(L(\frac12)\)	\(\approx\)	\(2.254716139\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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