LMFDB - L-function 16-966e8-1.1-c5e8-0-2

Dirichlet series

L(s) = 1

+ 32·2^-s + 72·3^-s + 576·4^-s + 25·5^-s + 2.30e3·6^-s − 392·7^-s + 7.68e3·8^-s + 2.91e3·9^-s + 800·10^-s + 417·11^-s + 4.14e4·12^-s + 417·13^-s − 1.25e4·14^-s + 1.80e3·15^-s + 8.44e4·16^-s + 1.36e3·17^-s + 9.33e4·18^-s + 2.99e3·19^-s + 1.44e4·20^-s − 2.82e4·21^-s + 1.33e4·22^-s + 4.23e3·23^-s + 5.52e5·24^-s − 9.75e3·25^-s + 1.33e4·26^-s + 8.74e4·27^-s − 2.25e5·28^-s + ⋯

L(s) = 1

+ 5.65·2^-s + 4.61·3^-s + 18·4^-s + 0.447·5^-s + 26.1·6^-s − 3.02·7^-s + 42.4·8^-s + 12·9^-s + 2.52·10^-s + 1.03·11^-s + 83.1·12^-s + 0.684·13^-s − 17.1·14^-s + 2.06·15^-s + 82.5·16^-s + 1.14·17^-s + 67.8·18^-s + 1.90·19^-s + 8.04·20^-s − 13.9·21^-s + 5.87·22^-s + 1.66·23^-s + 195.·24^-s − 3.12·25^-s + 3.87·26^-s + 23.0·27^-s − 54.4·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 23^{8}$
Sign:	$1$
Analytic conductor:	$3.31971\times 10^{17}$
Root analytic conductor:	$12.4471$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 23^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$184964.4036$
$L(\frac12)$	$\approx$	$184964.4036$
$L(\frac{7}{2})$		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 - p^{2} T )^{8} $
	3	$ ( 1 - p^{2} T )^{8} $
	7	$ ( 1 + p^{2} T )^{8} $
	23	$ ( 1 - p^{2} T )^{8} $
good	5	$ 1 - p^{2} T + 10377 T^{2} - 448618 T^{3} + 13403857 p T^{4} - 2929900059 T^{5} + 13120303559 p^{2} T^{6} - 12832693161698 T^{7} + 1172734342752212 T^{8} - 12832693161698 p^{5} T^{9} + 13120303559 p^{12} T^{10} - 2929900059 p^{15} T^{11} + 13403857 p^{21} T^{12} - 448618 p^{25} T^{13} + 10377 p^{30} T^{14} - p^{37} T^{15} + p^{40} T^{16} $
	11	$ 1 - 417 T + 587110 T^{2} - 241885351 T^{3} + 209581109293 T^{4} - 73451435042440 T^{5} + 52148924854242706 T^{6} - 16271844647130225922 T^{7} + $$94\!\cdots\!20$$ T^{8} - 16271844647130225922 p^{5} T^{9} + 52148924854242706 p^{10} T^{10} - 73451435042440 p^{15} T^{11} + 209581109293 p^{20} T^{12} - 241885351 p^{25} T^{13} + 587110 p^{30} T^{14} - 417 p^{35} T^{15} + p^{40} T^{16} $
	13	$ 1 - 417 T + 121871 p T^{2} - 363973890 T^{3} + 858799103807 T^{4} + 86357765917825 T^{5} + 129969600793026643 T^{6} + $$19\!\cdots\!10$$ T^{7} - $$16\!\cdots\!80$$ T^{8} + $$19\!\cdots\!10$$ p^{5} T^{9} + 129969600793026643 p^{10} T^{10} + 86357765917825 p^{15} T^{11} + 858799103807 p^{20} T^{12} - 363973890 p^{25} T^{13} + 121871 p^{31} T^{14} - 417 p^{35} T^{15} + p^{40} T^{16} $
	17	$ 1 - 1368 T + 5749586 T^{2} - 4710601024 T^{3} + 13073750197105 T^{4} - 4275007198510672 T^{5} + 16088090184245837902 T^{6} + $$46\!\cdots\!00$$ T^{7} + $$17\!\cdots\!32$$ T^{8} + $$46\!\cdots\!00$$ p^{5} T^{9} + 16088090184245837902 p^{10} T^{10} - 4275007198510672 p^{15} T^{11} + 13073750197105 p^{20} T^{12} - 4710601024 p^{25} T^{13} + 5749586 p^{30} T^{14} - 1368 p^{35} T^{15} + p^{40} T^{16} $
	19	$ 1 - 2999 T + 14161094 T^{2} - 30761082649 T^{3} + 84014349690045 T^{4} - 141932777966284696 T^{5} + $$29\!\cdots\!50$$ T^{6} - $$42\!\cdots\!10$$ T^{7} + $$78\!\cdots\!00$$ T^{8} - $$42\!\cdots\!10$$ p^{5} T^{9} + $$29\!\cdots\!50$$ p^{10} T^{10} - 141932777966284696 p^{15} T^{11} + 84014349690045 p^{20} T^{12} - 30761082649 p^{25} T^{13} + 14161094 p^{30} T^{14} - 2999 p^{35} T^{15} + p^{40} T^{16} $
	29	$ 1 - 4769 T + 91625968 T^{2} - 495266084049 T^{3} + 4857658992131293 T^{4} - 23235042588991975216 T^{5} + $$17\!\cdots\!22$$ T^{6} - $$24\!\cdots\!94$$ p T^{7} + $$41\!\cdots\!52$$ T^{8} - $$24\!\cdots\!94$$ p^{6} T^{9} + $$17\!\cdots\!22$$ p^{10} T^{10} - 23235042588991975216 p^{15} T^{11} + 4857658992131293 p^{20} T^{12} - 495266084049 p^{25} T^{13} + 91625968 p^{30} T^{14} - 4769 p^{35} T^{15} + p^{40} T^{16} $
	31	$ 1 - 4656 T + 26965802 T^{2} - 104126739332 T^{3} + 1048062152552553 T^{4} - 11670864482679066992 T^{5} + $$61\!\cdots\!66$$ T^{6} - $$24\!\cdots\!80$$ T^{7} + $$10\!\cdots\!36$$ T^{8} - $$24\!\cdots\!80$$ p^{5} T^{9} + $$61\!\cdots\!66$$ p^{10} T^{10} - 11670864482679066992 p^{15} T^{11} + 1048062152552553 p^{20} T^{12} - 104126739332 p^{25} T^{13} + 26965802 p^{30} T^{14} - 4656 p^{35} T^{15} + p^{40} T^{16} $
	37	$ 1 - 12289 T + 367904284 T^{2} - 4307889626053 T^{3} + 71974200329606137 T^{4} - $$71\!\cdots\!24$$ T^{5} + $$88\!\cdots\!10$$ T^{6} - $$74\!\cdots\!82$$ T^{7} + $$73\!\cdots\!76$$ T^{8} - $$74\!\cdots\!82$$ p^{5} T^{9} + $$88\!\cdots\!10$$ p^{10} T^{10} - $$71\!\cdots\!24$$ p^{15} T^{11} + 71974200329606137 p^{20} T^{12} - 4307889626053 p^{25} T^{13} + 367904284 p^{30} T^{14} - 12289 p^{35} T^{15} + p^{40} T^{16} $
	41	$ 1 - 4768 T + 585982429 T^{2} - 2323431561260 T^{3} + 171224134471420521 T^{4} - $$57\!\cdots\!60$$ T^{5} + $$32\!\cdots\!34$$ T^{6} - $$93\!\cdots\!48$$ T^{7} + $$43\!\cdots\!02$$ T^{8} - $$93\!\cdots\!48$$ p^{5} T^{9} + $$32\!\cdots\!34$$ p^{10} T^{10} - $$57\!\cdots\!60$$ p^{15} T^{11} + 171224134471420521 p^{20} T^{12} - 2323431561260 p^{25} T^{13} + 585982429 p^{30} T^{14} - 4768 p^{35} T^{15} + p^{40} T^{16} $
	43	$ 1 - 34975 T + 1145641249 T^{2} - 27042225365784 T^{3} + 564275372040848393 T^{4} - $$99\!\cdots\!11$$ T^{5} + $$15\!\cdots\!51$$ T^{6} - $$22\!\cdots\!26$$ T^{7} + $$28\!\cdots\!96$$ T^{8} - $$22\!\cdots\!26$$ p^{5} T^{9} + $$15\!\cdots\!51$$ p^{10} T^{10} - $$99\!\cdots\!11$$ p^{15} T^{11} + 564275372040848393 p^{20} T^{12} - 27042225365784 p^{25} T^{13} + 1145641249 p^{30} T^{14} - 34975 p^{35} T^{15} + p^{40} T^{16} $
	47	$ 1 - 16500 T + 1213937815 T^{2} - 21705286928044 T^{3} + 746478606038127350 T^{4} - $$12\!\cdots\!36$$ T^{5} + $$29\!\cdots\!85$$ T^{6} - $$43\!\cdots\!88$$ T^{7} + $$82\!\cdots\!98$$ T^{8} - $$43\!\cdots\!88$$ p^{5} T^{9} + $$29\!\cdots\!85$$ p^{10} T^{10} - $$12\!\cdots\!36$$ p^{15} T^{11} + 746478606038127350 p^{20} T^{12} - 21705286928044 p^{25} T^{13} + 1213937815 p^{30} T^{14} - 16500 p^{35} T^{15} + p^{40} T^{16} $
	53	$ 1 - 60747 T + 3923170693 T^{2} - 157023957872538 T^{3} + 6059711226271328265 T^{4} - $$18\!\cdots\!33$$ T^{5} + $$51\!\cdots\!87$$ T^{6} - $$12\!\cdots\!66$$ T^{7} + $$27\!\cdots\!32$$ T^{8} - $$12\!\cdots\!66$$ p^{5} T^{9} + $$51\!\cdots\!87$$ p^{10} T^{10} - $$18\!\cdots\!33$$ p^{15} T^{11} + 6059711226271328265 p^{20} T^{12} - 157023957872538 p^{25} T^{13} + 3923170693 p^{30} T^{14} - 60747 p^{35} T^{15} + p^{40} T^{16} $
	59	$ 1 - 50613 T + 4378330591 T^{2} - 160030114608544 T^{3} + 7495498513675883171 T^{4} - $$20\!\cdots\!17$$ T^{5} + $$71\!\cdots\!43$$ T^{6} - $$16\!\cdots\!46$$ T^{7} + $$52\!\cdots\!64$$ T^{8} - $$16\!\cdots\!46$$ p^{5} T^{9} + $$71\!\cdots\!43$$ p^{10} T^{10} - $$20\!\cdots\!17$$ p^{15} T^{11} + 7495498513675883171 p^{20} T^{12} - 160030114608544 p^{25} T^{13} + 4378330591 p^{30} T^{14} - 50613 p^{35} T^{15} + p^{40} T^{16} $
	61	$ 1 - 55269 T + 5349010003 T^{2} - 221463462475638 T^{3} + 12463194772267450495 T^{4} - $$40\!\cdots\!35$$ T^{5} + $$17\!\cdots\!75$$ T^{6} - $$47\!\cdots\!54$$ T^{7} + $$17\!\cdots\!24$$ T^{8} - $$47\!\cdots\!54$$ p^{5} T^{9} + $$17\!\cdots\!75$$ p^{10} T^{10} - $$40\!\cdots\!35$$ p^{15} T^{11} + 12463194772267450495 p^{20} T^{12} - 221463462475638 p^{25} T^{13} + 5349010003 p^{30} T^{14} - 55269 p^{35} T^{15} + p^{40} T^{16} $
	67	$ 1 - 84046 T + 10811387951 T^{2} - 658541433944148 T^{3} + 49267398885657393795 T^{4} - $$23\!\cdots\!76$$ T^{5} + $$12\!\cdots\!37$$ T^{6} - $$49\!\cdots\!54$$ T^{7} + $$21\!\cdots\!72$$ T^{8} - $$49\!\cdots\!54$$ p^{5} T^{9} + $$12\!\cdots\!37$$ p^{10} T^{10} - $$23\!\cdots\!76$$ p^{15} T^{11} + 49267398885657393795 p^{20} T^{12} - 658541433944148 p^{25} T^{13} + 10811387951 p^{30} T^{14} - 84046 p^{35} T^{15} + p^{40} T^{16} $
	71	$ 1 - 60760 T + 9094534379 T^{2} - 401265948044864 T^{3} + 36497992942580005451 T^{4} - $$13\!\cdots\!58$$ T^{5} + $$10\!\cdots\!41$$ T^{6} - $$33\!\cdots\!98$$ T^{7} + $$21\!\cdots\!36$$ T^{8} - $$33\!\cdots\!98$$ p^{5} T^{9} + $$10\!\cdots\!41$$ p^{10} T^{10} - $$13\!\cdots\!58$$ p^{15} T^{11} + 36497992942580005451 p^{20} T^{12} - 401265948044864 p^{25} T^{13} + 9094534379 p^{30} T^{14} - 60760 p^{35} T^{15} + p^{40} T^{16} $
	73	$ 1 - 97008 T + 9224855142 T^{2} - 722353685293132 T^{3} + 49155648698629925389 T^{4} - $$30\!\cdots\!84$$ T^{5} + $$16\!\cdots\!70$$ T^{6} - $$86\!\cdots\!68$$ T^{7} + $$41\!\cdots\!84$$ T^{8} - $$86\!\cdots\!68$$ p^{5} T^{9} + $$16\!\cdots\!70$$ p^{10} T^{10} - $$30\!\cdots\!84$$ p^{15} T^{11} + 49155648698629925389 p^{20} T^{12} - 722353685293132 p^{25} T^{13} + 9224855142 p^{30} T^{14} - 97008 p^{35} T^{15} + p^{40} T^{16} $
	79	$ 1 - 60570 T + 20702450286 T^{2} - 1163134940668850 T^{3} + $$19\!\cdots\!41$$ T^{4} - $$98\!\cdots\!36$$ T^{5} + $$11\!\cdots\!54$$ T^{6} - $$48\!\cdots\!56$$ T^{7} + $$42\!\cdots\!56$$ T^{8} - $$48\!\cdots\!56$$ p^{5} T^{9} + $$11\!\cdots\!54$$ p^{10} T^{10} - $$98\!\cdots\!36$$ p^{15} T^{11} + $$19\!\cdots\!41$$ p^{20} T^{12} - 1163134940668850 p^{25} T^{13} + 20702450286 p^{30} T^{14} - 60570 p^{35} T^{15} + p^{40} T^{16} $
	83	$ 1 - 217526 T + 42040470434 T^{2} - 5516603184179930 T^{3} + $$64\!\cdots\!25$$ T^{4} - $$61\!\cdots\!92$$ T^{5} + $$52\!\cdots\!14$$ T^{6} - $$39\!\cdots\!04$$ T^{7} + $$26\!\cdots\!88$$ T^{8} - $$39\!\cdots\!04$$ p^{5} T^{9} + $$52\!\cdots\!14$$ p^{10} T^{10} - $$61\!\cdots\!92$$ p^{15} T^{11} + $$64\!\cdots\!25$$ p^{20} T^{12} - 5516603184179930 p^{25} T^{13} + 42040470434 p^{30} T^{14} - 217526 p^{35} T^{15} + p^{40} T^{16} $
	89	$ 1 - 123662 T + 25576799231 T^{2} - 2554276048544792 T^{3} + $$30\!\cdots\!43$$ T^{4} - $$24\!\cdots\!68$$ T^{5} + $$22\!\cdots\!45$$ T^{6} - $$15\!\cdots\!82$$ T^{7} + $$13\!\cdots\!60$$ T^{8} - $$15\!\cdots\!82$$ p^{5} T^{9} + $$22\!\cdots\!45$$ p^{10} T^{10} - $$24\!\cdots\!68$$ p^{15} T^{11} + $$30\!\cdots\!43$$ p^{20} T^{12} - 2554276048544792 p^{25} T^{13} + 25576799231 p^{30} T^{14} - 123662 p^{35} T^{15} + p^{40} T^{16} $
	97	$ 1 - 234159 T + 68604511148 T^{2} - 10865905327445383 T^{3} + $$18\!\cdots\!13$$ T^{4} - $$22\!\cdots\!48$$ T^{5} + $$29\!\cdots\!74$$ T^{6} - $$29\!\cdots\!54$$ T^{7} + $$30\!\cdots\!68$$ T^{8} - $$29\!\cdots\!54$$ p^{5} T^{9} + $$29\!\cdots\!74$$ p^{10} T^{10} - $$22\!\cdots\!48$$ p^{15} T^{11} + $$18\!\cdots\!13$$ p^{20} T^{12} - 10865905327445383 p^{25} T^{13} + 68604511148 p^{30} T^{14} - 234159 p^{35} T^{15} + p^{40} T^{16} $
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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

−3.50476059197048597949280321367, −3.34922267478083548079232404356, −3.29210764272471999032740952950, −3.26144662310196191721228063912, −3.03126312386852922710494088773, −2.99656728283545592370257428163, −2.98531451262304957844014633837, −2.79926401197936699190421608223, −2.78087176930597757344322612725, −2.27473339425136845558731334980, −2.21760563570991726150087338439, −2.19899041382664532164392110721, −2.16138063722325810412071543539, −2.06867383825199929897216929666, −2.03651037908556629654463638124, −1.91283734602791617739083004449, −1.91257965719769744265893687641, −1.19049043200422413168506919877, −1.02315182736036750620293144194, −0.952040127132461507412769071793, −0.925024569968013861088821466717, −0.810206325882152570340577392220, −0.72534203372753880444180382793, −0.68062583630910161005285314620, −0.53824400039432670648701633094, 0.53824400039432670648701633094, 0.68062583630910161005285314620, 0.72534203372753880444180382793, 0.810206325882152570340577392220, 0.925024569968013861088821466717, 0.952040127132461507412769071793, 1.02315182736036750620293144194, 1.19049043200422413168506919877, 1.91257965719769744265893687641, 1.91283734602791617739083004449, 2.03651037908556629654463638124, 2.06867383825199929897216929666, 2.16138063722325810412071543539, 2.19899041382664532164392110721, 2.21760563570991726150087338439, 2.27473339425136845558731334980, 2.78087176930597757344322612725, 2.79926401197936699190421608223, 2.98531451262304957844014633837, 2.99656728283545592370257428163, 3.03126312386852922710494088773, 3.26144662310196191721228063912, 3.29210764272471999032740952950, 3.34922267478083548079232404356, 3.50476059197048597949280321367

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(184964.4036\)
\(L(\frac12)\)	\(\approx\)	\(184964.4036\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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