LMFDB - L-function 24-320e12-1.1-c5e12-0-1

Dirichlet series

L(s) = 1

+ 268·7^-s + 1.24e3·9^-s + 2.40e3·17^-s + 8.10e3·23^-s − 3.75e3·25^-s + 7.97e3·31^-s − 5.64e4·41^-s + 2.11e4·47^-s − 6.77e4·49^-s + 3.33e5·63^-s + 3.67e5·71^-s + 5.87e4·73^-s + 2.61e4·79^-s + 8.50e5·81^-s + 8.76e4·89^-s − 1.72e5·97^-s + 4.35e5·103^-s − 4.46e5·113^-s + 6.43e5·119^-s + 1.00e6·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 2.98e6·153^-s + ⋯

L(s) = 1

+ 2.06·7^-s + 5.11·9^-s + 2.01·17^-s + 3.19·23^-s − 6/5·25^-s + 1.49·31^-s − 5.24·41^-s + 1.39·47^-s − 4.02·49^-s + 10.5·63^-s + 8.65·71^-s + 1.29·73^-s + 0.472·79^-s + 14.4·81^-s + 1.17·89^-s − 1.85·97^-s + 4.04·103^-s − 3.28·113^-s + 4.16·119^-s + 6.22·121^-s + 5.50e−6·127^-s + 5.09e−6·131^-s + 4.55e−6·137^-s + 4.38e−6·139^-s + 3.69e−6·149^-s + 3.56e−6·151^-s + 10.3·153^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$2^{72} \cdot 5^{12}$
Sign:	$1$
Analytic conductor:	$3.33984\times 10^{20}$
Root analytic conductor:	$7.16399$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 2^{72} \cdot 5^{12} ,\ ( \ : [5/2]^{12} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$8.353851724$
$L(\frac12)$	$\approx$	$8.353851724$
$L(\frac{7}{2})$		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 $
	5	$ ( 1 + p^{4} T^{2} )^{6} $
good	3	$ 1 - 1244 T^{2} + 232282 p T^{4} - 230065580 T^{6} + 47479013407 T^{8} - 640563641560 p^{2} T^{10} + 7836105400132 p^{4} T^{12} - 640563641560 p^{12} T^{14} + 47479013407 p^{20} T^{16} - 230065580 p^{30} T^{18} + 232282 p^{41} T^{20} - 1244 p^{50} T^{22} + p^{60} T^{24} $
	7	$ ( 1 - 134 T + 60784 T^{2} - 9725794 T^{3} + 1872794971 T^{4} - 305623921700 T^{5} + 37504003751552 T^{6} - 305623921700 p^{5} T^{7} + 1872794971 p^{10} T^{8} - 9725794 p^{15} T^{9} + 60784 p^{20} T^{10} - 134 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	11	$ 1 - 1002172 T^{2} + 493219533810 T^{4} - 14825028358228548 p T^{6} + $$41\!\cdots\!27$$ T^{8} - $$86\!\cdots\!44$$ T^{10} + $$15\!\cdots\!36$$ T^{12} - $$86\!\cdots\!44$$ p^{10} T^{14} + $$41\!\cdots\!27$$ p^{20} T^{16} - 14825028358228548 p^{31} T^{18} + 493219533810 p^{40} T^{20} - 1002172 p^{50} T^{22} + p^{60} T^{24} $
	13	$ 1 - 2253844 T^{2} + 2609423635634 T^{4} - 2090949123359826660 T^{6} + $$12\!\cdots\!95$$ T^{8} - $$64\!\cdots\!44$$ T^{10} + $$26\!\cdots\!16$$ T^{12} - $$64\!\cdots\!44$$ p^{10} T^{14} + $$12\!\cdots\!95$$ p^{20} T^{16} - 2090949123359826660 p^{30} T^{18} + 2609423635634 p^{40} T^{20} - 2253844 p^{50} T^{22} + p^{60} T^{24} $
	17	$ ( 1 - 1200 T + 5829762 T^{2} - 7194221040 T^{3} + 17474250252687 T^{4} - 17877633531638880 T^{5} + 31833151505919656476 T^{6} - 17877633531638880 p^{5} T^{7} + 17474250252687 p^{10} T^{8} - 7194221040 p^{15} T^{9} + 5829762 p^{20} T^{10} - 1200 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	19	$ 1 - 1047044 p T^{2} + 197127587113970 T^{4} - $$12\!\cdots\!52$$ T^{6} + $$60\!\cdots\!07$$ T^{8} - $$21\!\cdots\!16$$ T^{10} + $$60\!\cdots\!52$$ T^{12} - $$21\!\cdots\!16$$ p^{10} T^{14} + $$60\!\cdots\!07$$ p^{20} T^{16} - $$12\!\cdots\!52$$ p^{30} T^{18} + 197127587113970 p^{40} T^{20} - 1047044 p^{51} T^{22} + p^{60} T^{24} $
	23	$ ( 1 - 4054 T + 24767560 T^{2} - 85869926290 T^{3} + 327319542066779 T^{4} - 869764424865766180 T^{5} + $$26\!\cdots\!28$$ T^{6} - 869764424865766180 p^{5} T^{7} + 327319542066779 p^{10} T^{8} - 85869926290 p^{15} T^{9} + 24767560 p^{20} T^{10} - 4054 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	29	$ 1 - 118766236 T^{2} + 8086517092300050 T^{4} - $$38\!\cdots\!40$$ T^{6} + $$13\!\cdots\!91$$ T^{8} - $$38\!\cdots\!40$$ T^{10} + $$88\!\cdots\!84$$ T^{12} - $$38\!\cdots\!40$$ p^{10} T^{14} + $$13\!\cdots\!91$$ p^{20} T^{16} - $$38\!\cdots\!40$$ p^{30} T^{18} + 8086517092300050 p^{40} T^{20} - 118766236 p^{50} T^{22} + p^{60} T^{24} $
	31	$ ( 1 - 3988 T + 3534574 p T^{2} - 302364450332 T^{3} + 5668865865566767 T^{4} - 12320621141215714888 T^{5} + $$19\!\cdots\!04$$ T^{6} - 12320621141215714888 p^{5} T^{7} + 5668865865566767 p^{10} T^{8} - 302364450332 p^{15} T^{9} + 3534574 p^{21} T^{10} - 3988 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	37	$ 1 - 298070820 T^{2} + 53469062376708978 T^{4} - $$69\!\cdots\!56$$ T^{6} + $$71\!\cdots\!03$$ T^{8} - $$61\!\cdots\!04$$ T^{10} + $$45\!\cdots\!00$$ T^{12} - $$61\!\cdots\!04$$ p^{10} T^{14} + $$71\!\cdots\!03$$ p^{20} T^{16} - $$69\!\cdots\!56$$ p^{30} T^{18} + 53469062376708978 p^{40} T^{20} - 298070820 p^{50} T^{22} + p^{60} T^{24} $
	41	$ ( 1 + 28204 T + 645547054 T^{2} + 9891921342268 T^{3} + 144344267066611295 T^{4} + $$17\!\cdots\!96$$ T^{5} + $$20\!\cdots\!08$$ T^{6} + $$17\!\cdots\!96$$ p^{5} T^{7} + 144344267066611295 p^{10} T^{8} + 9891921342268 p^{15} T^{9} + 645547054 p^{20} T^{10} + 28204 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	43	$ 1 - 854218268 T^{2} + 379569874740918830 T^{4} - $$11\!\cdots\!80$$ T^{6} + $$26\!\cdots\!71$$ T^{8} - $$50\!\cdots\!20$$ T^{10} + $$79\!\cdots\!96$$ T^{12} - $$50\!\cdots\!20$$ p^{10} T^{14} + $$26\!\cdots\!71$$ p^{20} T^{16} - $$11\!\cdots\!80$$ p^{30} T^{18} + 379569874740918830 p^{40} T^{20} - 854218268 p^{50} T^{22} + p^{60} T^{24} $
	47	$ ( 1 - 10586 T + 559351288 T^{2} - 141995130734 T^{3} + 85696824820019915 T^{4} + $$20\!\cdots\!52$$ T^{5} + $$41\!\cdots\!80$$ T^{6} + $$20\!\cdots\!52$$ p^{5} T^{7} + 85696824820019915 p^{10} T^{8} - 141995130734 p^{15} T^{9} + 559351288 p^{20} T^{10} - 10586 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	53	$ 1 - 2472111540 T^{2} + 3272109664551065298 T^{4} - $$29\!\cdots\!96$$ T^{6} + $$20\!\cdots\!23$$ T^{8} - $$11\!\cdots\!44$$ T^{10} + $$51\!\cdots\!20$$ T^{12} - $$11\!\cdots\!44$$ p^{10} T^{14} + $$20\!\cdots\!23$$ p^{20} T^{16} - $$29\!\cdots\!96$$ p^{30} T^{18} + 3272109664551065298 p^{40} T^{20} - 2472111540 p^{50} T^{22} + p^{60} T^{24} $
	59	$ 1 - 3285269100 T^{2} + 3759201793217414610 T^{4} - $$13\!\cdots\!44$$ T^{6} + $$12\!\cdots\!35$$ T^{8} - $$13\!\cdots\!20$$ T^{10} + $$52\!\cdots\!56$$ p^{2} T^{12} - $$13\!\cdots\!20$$ p^{10} T^{14} + $$12\!\cdots\!35$$ p^{20} T^{16} - $$13\!\cdots\!44$$ p^{30} T^{18} + 3759201793217414610 p^{40} T^{20} - 3285269100 p^{50} T^{22} + p^{60} T^{24} $
	61	$ 1 - 9275968036 T^{2} + 40037810273961512978 T^{4} - $$10\!\cdots\!60$$ T^{6} + $$19\!\cdots\!03$$ T^{8} - $$25\!\cdots\!00$$ T^{10} + $$24\!\cdots\!52$$ T^{12} - $$25\!\cdots\!00$$ p^{10} T^{14} + $$19\!\cdots\!03$$ p^{20} T^{16} - $$10\!\cdots\!60$$ p^{30} T^{18} + 40037810273961512978 p^{40} T^{20} - 9275968036 p^{50} T^{22} + p^{60} T^{24} $
	67	$ 1 - 7932747836 T^{2} + 31498024738605168590 T^{4} - $$85\!\cdots\!12$$ T^{6} + $$17\!\cdots\!55$$ T^{8} - $$30\!\cdots\!40$$ T^{10} + $$44\!\cdots\!12$$ T^{12} - $$30\!\cdots\!40$$ p^{10} T^{14} + $$17\!\cdots\!55$$ p^{20} T^{16} - $$85\!\cdots\!12$$ p^{30} T^{18} + 31498024738605168590 p^{40} T^{20} - 7932747836 p^{50} T^{22} + p^{60} T^{24} $
	71	$ ( 1 - 183852 T + 22719949218 T^{2} - 1935102444145668 T^{3} + $$13\!\cdots\!79$$ T^{4} - $$73\!\cdots\!52$$ T^{5} + $$34\!\cdots\!16$$ T^{6} - $$73\!\cdots\!52$$ p^{5} T^{7} + $$13\!\cdots\!79$$ p^{10} T^{8} - 1935102444145668 p^{15} T^{9} + 22719949218 p^{20} T^{10} - 183852 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	73	$ ( 1 - 29368 T + 61112786 p T^{2} - 258540669925656 T^{3} + 16168785780639245407 T^{4} - $$68\!\cdots\!32$$ T^{5} + $$45\!\cdots\!60$$ T^{6} - $$68\!\cdots\!32$$ p^{5} T^{7} + 16168785780639245407 p^{10} T^{8} - 258540669925656 p^{15} T^{9} + 61112786 p^{21} T^{10} - 29368 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	79	$ ( 1 - 13096 T + 12178064842 T^{2} + 46215677031496 T^{3} + 67636133741719889263 T^{4} + $$78\!\cdots\!08$$ T^{5} + $$24\!\cdots\!84$$ T^{6} + $$78\!\cdots\!08$$ p^{5} T^{7} + 67636133741719889263 p^{10} T^{8} + 46215677031496 p^{15} T^{9} + 12178064842 p^{20} T^{10} - 13096 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	83	$ 1 - 15441152812 T^{2} + $$13\!\cdots\!30$$ T^{4} - $$79\!\cdots\!24$$ T^{6} + $$37\!\cdots\!87$$ T^{8} - $$15\!\cdots\!88$$ T^{10} + $$63\!\cdots\!52$$ T^{12} - $$15\!\cdots\!88$$ p^{10} T^{14} + $$37\!\cdots\!87$$ p^{20} T^{16} - $$79\!\cdots\!24$$ p^{30} T^{18} + $$13\!\cdots\!30$$ p^{40} T^{20} - 15441152812 p^{50} T^{22} + p^{60} T^{24} $
	89	$ ( 1 - 43836 T + 13171715490 T^{2} - 211972966345068 T^{3} + 95738414377062864255 T^{4} - $$28\!\cdots\!20$$ T^{5} + $$69\!\cdots\!56$$ T^{6} - $$28\!\cdots\!20$$ p^{5} T^{7} + 95738414377062864255 p^{10} T^{8} - 211972966345068 p^{15} T^{9} + 13171715490 p^{20} T^{10} - 43836 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	97	$ ( 1 + 86168 T + 30789268338 T^{2} + 1787720936745144 T^{3} + $$45\!\cdots\!83$$ T^{4} + $$19\!\cdots\!32$$ T^{5} + $$44\!\cdots\!92$$ T^{6} + $$19\!\cdots\!32$$ p^{5} T^{7} + $$45\!\cdots\!83$$ p^{10} T^{8} + 1787720936745144 p^{15} T^{9} + 30789268338 p^{20} T^{10} + 86168 p^{25} T^{11} + p^{30} T^{12} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−2.98740234571001291945862075802, −2.96432725486218213637933293753, −2.85497455087869995988478517833, −2.62254517794785191316548404548, −2.33648850538947142255851319127, −2.32666534919635753934992337363, −2.22065827508355485272717699526, −2.00120589253765951849134538930, −1.96594322975377479839435599270, −1.95650767505735235070314941674, −1.95643298888955788323731527200, −1.72933792539219074705797510772, −1.58678953988749660784689739814, −1.51021097918693925359633144918, −1.21054183147506543412362221038, −1.16848387987432076999869294451, −1.13378599577057286660540389328, −1.09544435157951971165694200081, −1.03989998118181387591201969539, −0.880576596402362640833090949994, −0.75485689567826871008281629860, −0.61828648213851562301151332799, −0.42406494225146244070681855597, −0.23147502534738586381053198556, −0.04567840241889311487727436727, 0.04567840241889311487727436727, 0.23147502534738586381053198556, 0.42406494225146244070681855597, 0.61828648213851562301151332799, 0.75485689567826871008281629860, 0.880576596402362640833090949994, 1.03989998118181387591201969539, 1.09544435157951971165694200081, 1.13378599577057286660540389328, 1.16848387987432076999869294451, 1.21054183147506543412362221038, 1.51021097918693925359633144918, 1.58678953988749660784689739814, 1.72933792539219074705797510772, 1.95643298888955788323731527200, 1.95650767505735235070314941674, 1.96594322975377479839435599270, 2.00120589253765951849134538930, 2.22065827508355485272717699526, 2.32666534919635753934992337363, 2.33648850538947142255851319127, 2.62254517794785191316548404548, 2.85497455087869995988478517833, 2.96432725486218213637933293753, 2.98740234571001291945862075802

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(8.353851724\)
\(L(\frac12)\)	\(\approx\)	\(8.353851724\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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