LMFDB - L-function 24-15e24-1.1-c10e12-0-0

Dirichlet series

L(s) = 1

+ 4.70e3·4^-s − 5.21e4·7^-s + 2.76e5·13^-s + 1.00e7·16^-s + 1.89e6·19^-s − 2.45e8·28^-s − 1.21e8·31^-s − 1.01e8·37^-s + 1.55e8·43^-s − 7.23e8·49^-s + 1.30e9·52^-s + 2.94e8·61^-s + 1.28e10·64^-s − 5.07e9·67^-s − 2.78e9·73^-s + 8.92e9·76^-s − 4.71e9·79^-s − 1.44e10·91^-s + 2.51e10·97^-s + 2.68e10·103^-s + 2.86e9·109^-s − 5.22e11·112^-s + 1.77e11·121^-s − 5.71e11·124^-s + 127^-s + 131^-s − 9.89e10·133^-s + ⋯

L(s) = 1

+ 4.59·4^-s − 3.10·7^-s + 0.745·13^-s + 9.55·16^-s + 0.766·19^-s − 14.2·28^-s − 4.23·31^-s − 1.46·37^-s + 1.05·43^-s − 2.56·49^-s + 3.42·52^-s + 0.348·61^-s + 12.0·64^-s − 3.75·67^-s − 1.34·73^-s + 3.52·76^-s − 1.53·79^-s − 2.31·91^-s + 2.92·97^-s + 2.31·103^-s + 0.186·109^-s − 29.6·112^-s + 6.83·121^-s − 19.4·124^-s − 2.37·133^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+5)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$3^{24} \cdot 5^{24}$
Sign:	$1$
Analytic conductor:	$7.28460\times 10^{25}$
Root analytic conductor:	$11.9563$
Motivic weight:	$10$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 3^{24} \cdot 5^{24} ,\ ( \ : [5]^{12} ),\ 1 )$

Particular Values

$L(\frac{11}{2})$	$\approx$	$21.17464421$
$L(\frac12)$	$\approx$	$21.17464421$
$L(6)$		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 $
	5	$ 1 $
good	2	$ 1 - 2353 p T^{2} + 12130041 T^{4} - 356922295 p^{6} T^{6} + 66296541795 p^{9} T^{8} - 2591631199319 p^{14} T^{10} + 708816088715561 p^{16} T^{12} - 2591631199319 p^{34} T^{14} + 66296541795 p^{49} T^{16} - 356922295 p^{66} T^{18} + 12130041 p^{80} T^{20} - 2353 p^{101} T^{22} + p^{120} T^{24} $
	7	$ ( 1 + 26072 T + 1381305048 T^{2} + 4433473090408 p T^{3} + 17976769472844627 p^{2} T^{4} + 47289526066329566288 p^{3} T^{5} + $$13\!\cdots\!52$$ p^{4} T^{6} + 47289526066329566288 p^{13} T^{7} + 17976769472844627 p^{22} T^{8} + 4433473090408 p^{31} T^{9} + 1381305048 p^{40} T^{10} + 26072 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	11	$ 1 - 177177151784 T^{2} + $$15\!\cdots\!66$$ T^{4} - $$90\!\cdots\!20$$ T^{6} + $$39\!\cdots\!15$$ T^{8} - $$13\!\cdots\!04$$ T^{10} + $$38\!\cdots\!56$$ T^{12} - $$13\!\cdots\!04$$ p^{20} T^{14} + $$39\!\cdots\!15$$ p^{40} T^{16} - $$90\!\cdots\!20$$ p^{60} T^{18} + $$15\!\cdots\!66$$ p^{80} T^{20} - 177177151784 p^{100} T^{22} + p^{120} T^{24} $
	13	$ ( 1 - 138308 T + 435987712868 T^{2} + 19306563548398076 T^{3} + $$61\!\cdots\!63$$ T^{4} + $$21\!\cdots\!84$$ T^{5} + $$56\!\cdots\!92$$ T^{6} + $$21\!\cdots\!84$$ p^{10} T^{7} + $$61\!\cdots\!63$$ p^{20} T^{8} + 19306563548398076 p^{30} T^{9} + 435987712868 p^{40} T^{10} - 138308 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	17	$ 1 - 13539943740236 T^{2} + $$90\!\cdots\!26$$ T^{4} - $$24\!\cdots\!60$$ p T^{6} + $$13\!\cdots\!15$$ T^{8} - $$37\!\cdots\!56$$ T^{10} + $$83\!\cdots\!16$$ T^{12} - $$37\!\cdots\!56$$ p^{20} T^{14} + $$13\!\cdots\!15$$ p^{40} T^{16} - $$24\!\cdots\!60$$ p^{61} T^{18} + $$90\!\cdots\!26$$ p^{80} T^{20} - 13539943740236 p^{100} T^{22} + p^{120} T^{24} $
	19	$ ( 1 - 948640 T + 21229765294526 T^{2} - 23842053931646132000 T^{3} + $$20\!\cdots\!95$$ T^{4} - $$26\!\cdots\!00$$ T^{5} + $$13\!\cdots\!40$$ T^{6} - $$26\!\cdots\!00$$ p^{10} T^{7} + $$20\!\cdots\!95$$ p^{20} T^{8} - 23842053931646132000 p^{30} T^{9} + 21229765294526 p^{40} T^{10} - 948640 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	23	$ 1 - 226029510168420 T^{2} + $$27\!\cdots\!86$$ T^{4} - $$22\!\cdots\!00$$ T^{6} + $$14\!\cdots\!35$$ T^{8} - $$73\!\cdots\!00$$ T^{10} + $$32\!\cdots\!00$$ T^{12} - $$73\!\cdots\!00$$ p^{20} T^{14} + $$14\!\cdots\!35$$ p^{40} T^{16} - $$22\!\cdots\!00$$ p^{60} T^{18} + $$27\!\cdots\!86$$ p^{80} T^{20} - 226029510168420 p^{100} T^{22} + p^{120} T^{24} $
	29	$ 1 - 2386188356328500 T^{2} + $$28\!\cdots\!06$$ T^{4} - $$23\!\cdots\!00$$ T^{6} + $$15\!\cdots\!15$$ T^{8} - $$83\!\cdots\!00$$ T^{10} + $$37\!\cdots\!20$$ T^{12} - $$83\!\cdots\!00$$ p^{20} T^{14} + $$15\!\cdots\!15$$ p^{40} T^{16} - $$23\!\cdots\!00$$ p^{60} T^{18} + $$28\!\cdots\!06$$ p^{80} T^{20} - 2386188356328500 p^{100} T^{22} + p^{120} T^{24} $
	31	$ ( 1 + 60679444 T + 4294902756137226 T^{2} + $$16\!\cdots\!80$$ T^{3} + $$66\!\cdots\!15$$ T^{4} + $$18\!\cdots\!44$$ T^{5} + $$62\!\cdots\!36$$ T^{6} + $$18\!\cdots\!44$$ p^{10} T^{7} + $$66\!\cdots\!15$$ p^{20} T^{8} + $$16\!\cdots\!80$$ p^{30} T^{9} + 4294902756137226 p^{40} T^{10} + 60679444 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	37	$ ( 1 + 50888004 T + 14293378674677940 T^{2} + $$56\!\cdots\!28$$ T^{3} + $$11\!\cdots\!75$$ T^{4} + $$39\!\cdots\!40$$ T^{5} + $$66\!\cdots\!92$$ T^{6} + $$39\!\cdots\!40$$ p^{10} T^{7} + $$11\!\cdots\!75$$ p^{20} T^{8} + $$56\!\cdots\!28$$ p^{30} T^{9} + 14293378674677940 p^{40} T^{10} + 50888004 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	41	$ 1 - 87831691203055040 T^{2} + $$39\!\cdots\!86$$ T^{4} - $$12\!\cdots\!00$$ T^{6} + $$28\!\cdots\!35$$ T^{8} - $$52\!\cdots\!00$$ T^{10} + $$77\!\cdots\!00$$ T^{12} - $$52\!\cdots\!00$$ p^{20} T^{14} + $$28\!\cdots\!35$$ p^{40} T^{16} - $$12\!\cdots\!00$$ p^{60} T^{18} + $$39\!\cdots\!86$$ p^{80} T^{20} - 87831691203055040 p^{100} T^{22} + p^{120} T^{24} $
	43	$ ( 1 - 77888456 T + 83775798334453070 T^{2} - $$47\!\cdots\!72$$ T^{3} + $$35\!\cdots\!55$$ T^{4} - $$16\!\cdots\!00$$ T^{5} + $$94\!\cdots\!92$$ T^{6} - $$16\!\cdots\!00$$ p^{10} T^{7} + $$35\!\cdots\!55$$ p^{20} T^{8} - $$47\!\cdots\!72$$ p^{30} T^{9} + 83775798334453070 p^{40} T^{10} - 77888456 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	47	$ 1 + 23617465618972420 T^{2} + $$33\!\cdots\!86$$ T^{4} - $$10\!\cdots\!00$$ T^{6} + $$37\!\cdots\!35$$ T^{8} - $$69\!\cdots\!00$$ T^{10} + $$56\!\cdots\!00$$ T^{12} - $$69\!\cdots\!00$$ p^{20} T^{14} + $$37\!\cdots\!35$$ p^{40} T^{16} - $$10\!\cdots\!00$$ p^{60} T^{18} + $$33\!\cdots\!86$$ p^{80} T^{20} + 23617465618972420 p^{100} T^{22} + p^{120} T^{24} $
	53	$ 1 - 798387475246927100 T^{2} + $$38\!\cdots\!26$$ T^{4} - $$13\!\cdots\!00$$ T^{6} + $$36\!\cdots\!95$$ T^{8} - $$81\!\cdots\!00$$ T^{10} + $$15\!\cdots\!40$$ T^{12} - $$81\!\cdots\!00$$ p^{20} T^{14} + $$36\!\cdots\!95$$ p^{40} T^{16} - $$13\!\cdots\!00$$ p^{60} T^{18} + $$38\!\cdots\!26$$ p^{80} T^{20} - 798387475246927100 p^{100} T^{22} + p^{120} T^{24} $
	59	$ 1 - 1014582237425805224 T^{2} + $$79\!\cdots\!66$$ T^{4} - $$63\!\cdots\!20$$ T^{6} + $$46\!\cdots\!15$$ T^{8} - $$25\!\cdots\!04$$ T^{10} + $$13\!\cdots\!16$$ T^{12} - $$25\!\cdots\!04$$ p^{20} T^{14} + $$46\!\cdots\!15$$ p^{40} T^{16} - $$63\!\cdots\!20$$ p^{60} T^{18} + $$79\!\cdots\!66$$ p^{80} T^{20} - 1014582237425805224 p^{100} T^{22} + p^{120} T^{24} $
	61	$ ( 1 - 147192388 T + 3190256525173143386 T^{2} - $$40\!\cdots\!20$$ T^{3} + $$47\!\cdots\!55$$ T^{4} - $$52\!\cdots\!88$$ T^{5} + $$42\!\cdots\!24$$ T^{6} - $$52\!\cdots\!88$$ p^{10} T^{7} + $$47\!\cdots\!55$$ p^{20} T^{8} - $$40\!\cdots\!20$$ p^{30} T^{9} + 3190256525173143386 p^{40} T^{10} - 147192388 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	67	$ ( 1 + 2537593976 T + 9448468876581813990 T^{2} + $$17\!\cdots\!52$$ T^{3} + $$37\!\cdots\!35$$ T^{4} + $$55\!\cdots\!20$$ T^{5} + $$87\!\cdots\!32$$ T^{6} + $$55\!\cdots\!20$$ p^{10} T^{7} + $$37\!\cdots\!35$$ p^{20} T^{8} + $$17\!\cdots\!52$$ p^{30} T^{9} + 9448468876581813990 p^{40} T^{10} + 2537593976 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	71	$ 1 - 21927496318634330540 T^{2} + $$20\!\cdots\!26$$ T^{4} - $$10\!\cdots\!00$$ T^{6} + $$25\!\cdots\!95$$ T^{8} - $$42\!\cdots\!00$$ T^{10} - $$13\!\cdots\!60$$ T^{12} - $$42\!\cdots\!00$$ p^{20} T^{14} + $$25\!\cdots\!95$$ p^{40} T^{16} - $$10\!\cdots\!00$$ p^{60} T^{18} + $$20\!\cdots\!26$$ p^{80} T^{20} - 21927496318634330540 p^{100} T^{22} + p^{120} T^{24} $
	73	$ ( 1 + 1394387332 T + 18937045779026394698 T^{2} + $$26\!\cdots\!36$$ T^{3} + $$17\!\cdots\!23$$ T^{4} + $$20\!\cdots\!04$$ T^{5} + $$95\!\cdots\!52$$ T^{6} + $$20\!\cdots\!04$$ p^{10} T^{7} + $$17\!\cdots\!23$$ p^{20} T^{8} + $$26\!\cdots\!36$$ p^{30} T^{9} + 18937045779026394698 p^{40} T^{10} + 1394387332 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	79	$ ( 1 + 2355487444 T + 21527905744104060426 T^{2} + $$45\!\cdots\!80$$ T^{3} + $$21\!\cdots\!15$$ T^{4} + $$13\!\cdots\!44$$ T^{5} + $$18\!\cdots\!36$$ T^{6} + $$13\!\cdots\!44$$ p^{10} T^{7} + $$21\!\cdots\!15$$ p^{20} T^{8} + $$45\!\cdots\!80$$ p^{30} T^{9} + 21527905744104060426 p^{40} T^{10} + 2355487444 p^{50} T^{11} + p^{60} T^{12} )^{2} $
	83	$ 1 - 85571656789946386620 T^{2} + $$43\!\cdots\!86$$ T^{4} - $$15\!\cdots\!00$$ T^{6} + $$40\!\cdots\!35$$ T^{8} - $$86\!\cdots\!00$$ T^{10} + $$14\!\cdots\!00$$ T^{12} - $$86\!\cdots\!00$$ p^{20} T^{14} + $$40\!\cdots\!35$$ p^{40} T^{16} - $$15\!\cdots\!00$$ p^{60} T^{18} + $$43\!\cdots\!86$$ p^{80} T^{20} - 85571656789946386620 p^{100} T^{22} + p^{120} T^{24} $
	89	$ 1 - 52539612244756358304 T^{2} + $$36\!\cdots\!66$$ T^{4} - $$16\!\cdots\!20$$ T^{6} + $$71\!\cdots\!15$$ T^{8} - $$24\!\cdots\!04$$ T^{10} + $$86\!\cdots\!36$$ T^{12} - $$24\!\cdots\!04$$ p^{20} T^{14} + $$71\!\cdots\!15$$ p^{40} T^{16} - $$16\!\cdots\!20$$ p^{60} T^{18} + $$36\!\cdots\!66$$ p^{80} T^{20} - 52539612244756358304 p^{100} T^{22} + p^{120} T^{24} $
	97	$ ( 1 - 12557923164 T + $$28\!\cdots\!70$$ T^{2} - $$24\!\cdots\!28$$ T^{3} + $$31\!\cdots\!75$$ T^{4} - $$20\!\cdots\!80$$ T^{5} + $$23\!\cdots\!72$$ T^{6} - $$20\!\cdots\!80$$ p^{10} T^{7} + $$31\!\cdots\!75$$ p^{20} T^{8} - $$24\!\cdots\!28$$ p^{30} T^{9} + $$28\!\cdots\!70$$ p^{40} T^{10} - 12557923164 p^{50} T^{11} + p^{60} 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

−2.73736753778797568129978071486, −2.43740052717937955540898663199, −2.33914997170433832582345139039, −2.07555788177703231028223764349, −1.90714938920162430985465527697, −1.90442969394160002649406604485, −1.87911559081899381011338437444, −1.87404065672850506613269648076, −1.80633310809129597429708633346, −1.75787291444263541605961035625, −1.69018128520024487680827063127, −1.67889327211531914557417756383, −1.56268516754701185220108264404, −1.28708166717013648985344021712, −1.25162240295439135611866139510, −0.936571386819965092064327225101, −0.804606899280989596316865587409, −0.73483853528480367044020444271, −0.66205472713089472741536124087, −0.65647519884527180275800435485, −0.47578702337513975141856671014, −0.35480960356893431664266728565, −0.33398732491416712520493411571, −0.15408245959714151059939510006, −0.10798066949398747323381948624, 0.10798066949398747323381948624, 0.15408245959714151059939510006, 0.33398732491416712520493411571, 0.35480960356893431664266728565, 0.47578702337513975141856671014, 0.65647519884527180275800435485, 0.66205472713089472741536124087, 0.73483853528480367044020444271, 0.804606899280989596316865587409, 0.936571386819965092064327225101, 1.25162240295439135611866139510, 1.28708166717013648985344021712, 1.56268516754701185220108264404, 1.67889327211531914557417756383, 1.69018128520024487680827063127, 1.75787291444263541605961035625, 1.80633310809129597429708633346, 1.87404065672850506613269648076, 1.87911559081899381011338437444, 1.90442969394160002649406604485, 1.90714938920162430985465527697, 2.07555788177703231028223764349, 2.33914997170433832582345139039, 2.43740052717937955540898663199, 2.73736753778797568129978071486

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 $\F_{q}$

\(L(\frac{11}{2})\)	\(\approx\)	\(21.17464421\)
\(L(\frac12)\)	\(\approx\)	\(21.17464421\)
\(L(6)\)		not available
\(L(1)\)		not available

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