LMFDB - L-function 24-63e12-1.1-c5e12-0-0

Dirichlet series

L(s) = 1

+ 5·4^-s + 142·7^-s + 308·13^-s + 994·16^-s + 9.42e3·19^-s + 5.61e3·25^-s + 710·28^-s + 2.34e4·31^-s − 1.81e4·37^-s − 8.73e4·43^-s + 2.52e4·49^-s + 1.54e3·52^-s − 1.61e4·61^-s + 4.81e4·64^-s + 1.44e5·67^-s − 1.00e5·73^-s + 4.71e4·76^-s + 1.01e5·79^-s + 4.37e4·91^-s − 8.66e5·97^-s + 2.80e4·100^-s + 1.45e5·103^-s + 2.43e5·109^-s + 1.41e5·112^-s + 6.37e4·121^-s + 1.17e5·124^-s + 127^-s + ⋯

L(s) = 1

+ 5/32·4^-s + 1.09·7^-s + 0.505·13^-s + 0.970·16^-s + 5.98·19^-s + 1.79·25^-s + 0.171·28^-s + 4.37·31^-s − 2.18·37^-s − 7.20·43^-s + 1.50·49^-s + 0.0789·52^-s − 0.555·61^-s + 1.46·64^-s + 3.93·67^-s − 2.19·73^-s + 0.935·76^-s + 1.83·79^-s + 0.553·91^-s − 9.34·97^-s + 0.280·100^-s + 1.34·103^-s + 1.96·109^-s + 1.06·112^-s + 0.395·121^-s + 0.683·124^-s + 5.50e−6·127^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$3^{24} \cdot 7^{12}$
Sign:	$1$
Analytic conductor:	$1.13243\times 10^{12}$
Root analytic conductor:	$3.17870$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 3^{24} \cdot 7^{12} ,\ ( \ : [5/2]^{12} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$25.78670721$
$L(\frac12)$	$\approx$	$25.78670721$
$L(\frac{7}{2})$		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 $
	7	$ ( 1 - 71 T - 724 p T^{2} + 6691 p^{3} T^{3} - 724 p^{6} T^{4} - 71 p^{10} T^{5} + p^{15} T^{6} )^{2} $
good	2	$ 1 - 5 T^{2} - 969 T^{4} - 19149 p T^{6} + 13375 p^{2} T^{8} + 1343731 p^{4} T^{10} + 21675145 p^{6} T^{12} + 1343731 p^{14} T^{14} + 13375 p^{22} T^{16} - 19149 p^{31} T^{18} - 969 p^{40} T^{20} - 5 p^{50} T^{22} + p^{60} T^{24} $
	5	$ 1 - 5612 T^{2} - 2201904 T^{4} + 22496200452 T^{6} + 212632262258536 T^{8} - 406850851265056172 T^{10} - $$10\!\cdots\!54$$ T^{12} - 406850851265056172 p^{10} T^{14} + 212632262258536 p^{20} T^{16} + 22496200452 p^{30} T^{18} - 2201904 p^{40} T^{20} - 5612 p^{50} T^{22} + p^{60} T^{24} $
	11	$ 1 - 63728 T^{2} - 38186321040 T^{4} + 367330434605964 T^{6} + $$59\!\cdots\!48$$ T^{8} + $$32\!\cdots\!84$$ T^{10} - $$11\!\cdots\!34$$ T^{12} + $$32\!\cdots\!84$$ p^{10} T^{14} + $$59\!\cdots\!48$$ p^{20} T^{16} + 367330434605964 p^{30} T^{18} - 38186321040 p^{40} T^{20} - 63728 p^{50} T^{22} + p^{60} T^{24} $
	13	$ ( 1 - 77 T + 329903 T^{2} + 2901514 T^{3} + 329903 p^{5} T^{4} - 77 p^{10} T^{5} + p^{15} T^{6} )^{4} $
	17	$ 1 - 366750 p T^{2} + 20433382630821 T^{4} - 49353060269357285818 T^{6} + $$98\!\cdots\!70$$ T^{8} - $$16\!\cdots\!14$$ T^{10} + $$24\!\cdots\!45$$ T^{12} - $$16\!\cdots\!14$$ p^{10} T^{14} + $$98\!\cdots\!70$$ p^{20} T^{16} - 49353060269357285818 p^{30} T^{18} + 20433382630821 p^{40} T^{20} - 366750 p^{51} T^{22} + p^{60} T^{24} $
	19	$ ( 1 - 4711 T + 7468820 T^{2} - 15280705839 T^{3} + 52954224290656 T^{4} - 4398376818264401 p T^{5} + 245964098840846134 p^{2} T^{6} - 4398376818264401 p^{6} T^{7} + 52954224290656 p^{10} T^{8} - 15280705839 p^{15} T^{9} + 7468820 p^{20} T^{10} - 4711 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	23	$ 1 - 28426626 T^{2} + 428818666979349 T^{4} - $$46\!\cdots\!94$$ T^{6} + $$41\!\cdots\!02$$ T^{8} - $$31\!\cdots\!38$$ T^{10} + $$20\!\cdots\!17$$ T^{12} - $$31\!\cdots\!38$$ p^{10} T^{14} + $$41\!\cdots\!02$$ p^{20} T^{16} - $$46\!\cdots\!94$$ p^{30} T^{18} + 428818666979349 p^{40} T^{20} - 28426626 p^{50} T^{22} + p^{60} T^{24} $
	29	$ ( 1 + 56877260 T^{2} + 2153523833778304 T^{4} + $$50\!\cdots\!14$$ T^{6} + 2153523833778304 p^{10} T^{8} + 56877260 p^{20} T^{10} + p^{30} T^{12} )^{2} $
	31	$ ( 1 - 11711 T + 45990357 T^{2} + 101948632338 T^{3} - 1148073413015249 T^{4} - 1776754603256154347 T^{5} + $$38\!\cdots\!26$$ T^{6} - 1776754603256154347 p^{5} T^{7} - 1148073413015249 p^{10} T^{8} + 101948632338 p^{15} T^{9} + 45990357 p^{20} T^{10} - 11711 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	37	$ ( 1 + 9091 T - 48844230 T^{2} - 1261544169639 T^{3} - 2996267741214776 T^{4} + 39461171991303043615 T^{5} + $$57\!\cdots\!20$$ T^{6} + 39461171991303043615 p^{5} T^{7} - 2996267741214776 p^{10} T^{8} - 1261544169639 p^{15} T^{9} - 48844230 p^{20} T^{10} + 9091 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	41	$ ( 1 - 32763274 T^{2} + 36122988699522367 T^{4} - $$94\!\cdots\!80$$ T^{6} + 36122988699522367 p^{10} T^{8} - 32763274 p^{20} T^{10} + p^{30} T^{12} )^{2} $
	43	$ ( 1 + 21843 T + 434197953 T^{2} + 4778588910994 T^{3} + 434197953 p^{5} T^{4} + 21843 p^{10} T^{5} + p^{15} T^{6} )^{4} $
	47	$ 1 - 762615162 T^{2} + 276461698947460533 T^{4} - $$68\!\cdots\!82$$ T^{6} + $$14\!\cdots\!82$$ T^{8} - $$26\!\cdots\!26$$ T^{10} + $$48\!\cdots\!25$$ T^{12} - $$26\!\cdots\!26$$ p^{10} T^{14} + $$14\!\cdots\!82$$ p^{20} T^{16} - $$68\!\cdots\!82$$ p^{30} T^{18} + 276461698947460533 p^{40} T^{20} - 762615162 p^{50} T^{22} + p^{60} T^{24} $
	53	$ 1 - 1842711660 T^{2} + 1873007141082901776 T^{4} - $$12\!\cdots\!72$$ T^{6} + $$62\!\cdots\!60$$ T^{8} - $$25\!\cdots\!56$$ T^{10} + $$10\!\cdots\!30$$ T^{12} - $$25\!\cdots\!56$$ p^{10} T^{14} + $$62\!\cdots\!60$$ p^{20} T^{16} - $$12\!\cdots\!72$$ p^{30} T^{18} + 1873007141082901776 p^{40} T^{20} - 1842711660 p^{50} T^{22} + p^{60} T^{24} $
	59	$ 1 + 405776592 T^{2} - 1033610905572789456 T^{4} - $$62\!\cdots\!84$$ T^{6} + $$71\!\cdots\!76$$ T^{8} - $$62\!\cdots\!48$$ T^{10} - $$43\!\cdots\!18$$ T^{12} - $$62\!\cdots\!48$$ p^{10} T^{14} + $$71\!\cdots\!76$$ p^{20} T^{16} - $$62\!\cdots\!84$$ p^{30} T^{18} - 1033610905572789456 p^{40} T^{20} + 405776592 p^{50} T^{22} + p^{60} T^{24} $
	61	$ ( 1 + 8078 T - 1972526323 T^{2} - 137138925462 T^{3} + 2364406542104704030 T^{4} - $$51\!\cdots\!86$$ T^{5} - $$22\!\cdots\!15$$ T^{6} - $$51\!\cdots\!86$$ p^{5} T^{7} + 2364406542104704030 p^{10} T^{8} - 137138925462 p^{15} T^{9} - 1972526323 p^{20} T^{10} + 8078 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	67	$ ( 1 - 72325 T - 323604628 T^{2} + 3086336722323 T^{3} + 8714131286027959888 T^{4} - $$17\!\cdots\!97$$ T^{5} - $$52\!\cdots\!26$$ T^{6} - $$17\!\cdots\!97$$ p^{5} T^{7} + 8714131286027959888 p^{10} T^{8} + 3086336722323 p^{15} T^{9} - 323604628 p^{20} T^{10} - 72325 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	71	$ ( 1 + 7343984298 T^{2} + 25986157270151134623 T^{4} + $$57\!\cdots\!08$$ T^{6} + 25986157270151134623 p^{10} T^{8} + 7343984298 p^{20} T^{10} + p^{30} T^{12} )^{2} $
	73	$ ( 1 + 50029 T - 4199541354 T^{2} - 81413963652357 T^{3} + 20603785162045774216 T^{4} + $$22\!\cdots\!61$$ T^{5} - $$42\!\cdots\!84$$ T^{6} + $$22\!\cdots\!61$$ p^{5} T^{7} + 20603785162045774216 p^{10} T^{8} - 81413963652357 p^{15} T^{9} - 4199541354 p^{20} T^{10} + 50029 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	79	$ ( 1 - 50997 T - 3066101115 T^{2} + 115573939091150 T^{3} + 4359756628739682567 T^{4} + $$14\!\cdots\!11$$ T^{5} - $$24\!\cdots\!14$$ T^{6} + $$14\!\cdots\!11$$ p^{5} T^{7} + 4359756628739682567 p^{10} T^{8} + 115573939091150 p^{15} T^{9} - 3066101115 p^{20} T^{10} - 50997 p^{25} T^{11} + p^{30} T^{12} )^{2} $
	83	$ ( 1 + 15907224432 T^{2} + $$11\!\cdots\!44$$ T^{4} + $$52\!\cdots\!98$$ T^{6} + $$11\!\cdots\!44$$ p^{10} T^{8} + 15907224432 p^{20} T^{10} + p^{30} T^{12} )^{2} $
	89	$ 1 - 30683976398 T^{2} + $$53\!\cdots\!01$$ T^{4} - $$65\!\cdots\!94$$ T^{6} + $$61\!\cdots\!06$$ T^{8} - $$46\!\cdots\!10$$ T^{10} + $$28\!\cdots\!89$$ T^{12} - $$46\!\cdots\!10$$ p^{10} T^{14} + $$61\!\cdots\!06$$ p^{20} T^{16} - $$65\!\cdots\!94$$ p^{30} T^{18} + $$53\!\cdots\!01$$ p^{40} T^{20} - 30683976398 p^{50} T^{22} + p^{60} T^{24} $
	97	$ ( 1 + 216524 T + 31765510816 T^{2} + 3171927842758234 T^{3} + 31765510816 p^{5} T^{4} + 216524 p^{10} T^{5} + p^{15} T^{6} )^{4} $
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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

−4.51191873812479901716679137682, −4.45451588099477670492902268806, −3.90166697080233916253372226402, −3.78453504970872479311715876993, −3.76952127395953585647105708599, −3.63249163008738492953872051425, −3.46680357121096204315094465715, −3.40359457452569665556106217238, −3.37438777597830535738007355923, −2.89974664570084167583647620332, −2.77364839949014680663238920139, −2.75374132522611519442951574506, −2.68990965412730396431462304738, −2.62174599366163637751140324053, −2.25771672610148598719412816747, −1.66693094890332641188782487930, −1.61384001157345353117703984267, −1.59873024521014859638442577510, −1.31221771038398314558836425364, −1.25811851216410797731156737387, −1.16674782002206560927525559365, −0.815280316097173741732453913993, −0.77828172390399817671959658371, −0.35040597774110444556681253239, −0.28655212983406735446375631174, 0.28655212983406735446375631174, 0.35040597774110444556681253239, 0.77828172390399817671959658371, 0.815280316097173741732453913993, 1.16674782002206560927525559365, 1.25811851216410797731156737387, 1.31221771038398314558836425364, 1.59873024521014859638442577510, 1.61384001157345353117703984267, 1.66693094890332641188782487930, 2.25771672610148598719412816747, 2.62174599366163637751140324053, 2.68990965412730396431462304738, 2.75374132522611519442951574506, 2.77364839949014680663238920139, 2.89974664570084167583647620332, 3.37438777597830535738007355923, 3.40359457452569665556106217238, 3.46680357121096204315094465715, 3.63249163008738492953872051425, 3.76952127395953585647105708599, 3.78453504970872479311715876993, 3.90166697080233916253372226402, 4.45451588099477670492902268806, 4.51191873812479901716679137682

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(25.78670721\)
\(L(\frac12)\)	\(\approx\)	\(25.78670721\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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