LMFDB - L-function 24-208e12-1.1-c11e12-0-1

Dirichlet series

L(s) = 1

+ 488·3^-s − 6.16e5·9^-s + 3.20e6·13^-s + 1.21e7·17^-s − 5.81e6·23^-s + 2.96e8·25^-s − 3.50e8·27^-s − 2.44e8·29^-s + 1.56e9·39^-s − 2.29e9·43^-s + 1.00e10·49^-s + 5.95e9·51^-s − 4.60e9·53^-s − 1.37e10·61^-s − 2.83e9·69^-s + 1.44e11·75^-s − 1.80e10·79^-s + 1.17e11·81^-s − 1.19e11·87^-s + 4.87e11·101^-s + 1.71e10·103^-s + 2.82e11·107^-s − 1.06e12·113^-s − 1.97e12·117^-s + 1.42e12·121^-s + 127^-s − 1.11e12·129^-s + ⋯

L(s) = 1

+ 1.15·3^-s − 3.48·9^-s + 2.39·13^-s + 2.08·17^-s − 0.188·23^-s + 6.06·25^-s − 4.70·27^-s − 2.21·29^-s + 2.77·39^-s − 2.38·43^-s + 5.09·49^-s + 2.41·51^-s − 1.51·53^-s − 2.08·61^-s − 0.218·69^-s + 7.02·75^-s − 0.659·79^-s + 3.74·81^-s − 2.56·87^-s + 4.61·101^-s + 0.145·103^-s + 1.94·107^-s − 5.46·113^-s − 8.34·117^-s + 5.00·121^-s − 2.75·129^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$2^{48} \cdot 13^{12}$
Sign:	$1$
Analytic conductor:	$2.77602\times 10^{26}$
Root analytic conductor:	$12.6418$
Motivic weight:	$11$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 2^{48} \cdot 13^{12} ,\ ( \ : [11/2]^{12} ),\ 1 )$

Particular Values

$L(6)$	$\approx$	$46.24772325$
$L(\frac12)$	$\approx$	$46.24772325$
$L(\frac{13}{2})$		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 $
	13	$ 1 - 246836 p T + 336416330 p^{3} T^{2} + 33972109021932 p^{5} T^{3} - 289815087133718528 p^{7} T^{4} - 86095928318880858112 p^{10} T^{5} + $$97\!\cdots\!80$$ p^{15} T^{6} - 86095928318880858112 p^{21} T^{7} - 289815087133718528 p^{29} T^{8} + 33972109021932 p^{38} T^{9} + 336416330 p^{47} T^{10} - 246836 p^{56} T^{11} + p^{66} T^{12} $
good	3	$ ( 1 - 244 T + 1636 p^{5} T^{2} - 9613376 p^{2} T^{3} + 1338661844 p^{4} T^{4} - 30477119284 p^{6} T^{5} + 3708080134390 p^{8} T^{6} - 30477119284 p^{17} T^{7} + 1338661844 p^{26} T^{8} - 9613376 p^{35} T^{9} + 1636 p^{49} T^{10} - 244 p^{55} T^{11} + p^{66} T^{12} )^{2} $
	5	$ 1 - 296019944 T^{2} + 1924876342772288 p^{2} T^{4} - $$86\!\cdots\!48$$ p^{4} T^{6} + $$29\!\cdots\!88$$ p^{6} T^{8} - $$79\!\cdots\!08$$ p^{8} T^{10} + $$17\!\cdots\!74$$ p^{10} T^{12} - $$79\!\cdots\!08$$ p^{30} T^{14} + $$29\!\cdots\!88$$ p^{50} T^{16} - $$86\!\cdots\!48$$ p^{70} T^{18} + 1924876342772288 p^{90} T^{20} - 296019944 p^{110} T^{22} + p^{132} T^{24} $
	7	$ 1 - 1439593080 p T^{2} + 7513387543617322488 p T^{4} - $$17\!\cdots\!60$$ T^{6} + $$91\!\cdots\!04$$ p^{2} T^{8} - $$38\!\cdots\!92$$ p^{4} T^{10} + $$15\!\cdots\!26$$ p^{6} T^{12} - $$38\!\cdots\!92$$ p^{26} T^{14} + $$91\!\cdots\!04$$ p^{46} T^{16} - $$17\!\cdots\!60$$ p^{66} T^{18} + 7513387543617322488 p^{89} T^{20} - 1439593080 p^{111} T^{22} + p^{132} T^{24} $
	11	$ 1 - 1429047020516 T^{2} + $$11\!\cdots\!30$$ T^{4} - $$68\!\cdots\!16$$ T^{6} + $$31\!\cdots\!63$$ T^{8} - $$11\!\cdots\!08$$ T^{10} + $$36\!\cdots\!72$$ T^{12} - $$11\!\cdots\!08$$ p^{22} T^{14} + $$31\!\cdots\!63$$ p^{44} T^{16} - $$68\!\cdots\!16$$ p^{66} T^{18} + $$11\!\cdots\!30$$ p^{88} T^{20} - 1429047020516 p^{110} T^{22} + p^{132} T^{24} $
	17	$ ( 1 - 6099384 T + 65714447623728 T^{2} - $$60\!\cdots\!64$$ T^{3} + $$40\!\cdots\!32$$ T^{4} - $$25\!\cdots\!56$$ T^{5} + $$19\!\cdots\!02$$ T^{6} - $$25\!\cdots\!56$$ p^{11} T^{7} + $$40\!\cdots\!32$$ p^{22} T^{8} - $$60\!\cdots\!64$$ p^{33} T^{9} + 65714447623728 p^{44} T^{10} - 6099384 p^{55} T^{11} + p^{66} T^{12} )^{2} $
	19	$ 1 - 554101967186196 T^{2} + $$19\!\cdots\!22$$ T^{4} - $$46\!\cdots\!60$$ T^{6} + $$89\!\cdots\!75$$ T^{8} - $$13\!\cdots\!28$$ T^{10} + $$17\!\cdots\!12$$ T^{12} - $$13\!\cdots\!28$$ p^{22} T^{14} + $$89\!\cdots\!75$$ p^{44} T^{16} - $$46\!\cdots\!60$$ p^{66} T^{18} + $$19\!\cdots\!22$$ p^{88} T^{20} - 554101967186196 p^{110} T^{22} + p^{132} T^{24} $
	23	$ ( 1 + 2905296 T + 4052545647748770 T^{2} - $$11\!\cdots\!44$$ T^{3} + $$72\!\cdots\!75$$ T^{4} - $$45\!\cdots\!44$$ T^{5} + $$82\!\cdots\!80$$ T^{6} - $$45\!\cdots\!44$$ p^{11} T^{7} + $$72\!\cdots\!75$$ p^{22} T^{8} - $$11\!\cdots\!44$$ p^{33} T^{9} + 4052545647748770 p^{44} T^{10} + 2905296 p^{55} T^{11} + p^{66} T^{12} )^{2} $
	29	$ ( 1 + 122231556 T + 52401806364164466 T^{2} + $$68\!\cdots\!76$$ T^{3} + $$13\!\cdots\!55$$ T^{4} + $$15\!\cdots\!44$$ T^{5} + $$20\!\cdots\!96$$ T^{6} + $$15\!\cdots\!44$$ p^{11} T^{7} + $$13\!\cdots\!55$$ p^{22} T^{8} + $$68\!\cdots\!76$$ p^{33} T^{9} + 52401806364164466 p^{44} T^{10} + 122231556 p^{55} T^{11} + p^{66} T^{12} )^{2} $
	31	$ 1 - 157944828788449044 T^{2} + $$12\!\cdots\!18$$ T^{4} - $$67\!\cdots\!16$$ T^{6} + $$27\!\cdots\!67$$ T^{8} - $$91\!\cdots\!92$$ T^{10} + $$25\!\cdots\!12$$ T^{12} - $$91\!\cdots\!92$$ p^{22} T^{14} + $$27\!\cdots\!67$$ p^{44} T^{16} - $$67\!\cdots\!16$$ p^{66} T^{18} + $$12\!\cdots\!18$$ p^{88} T^{20} - 157944828788449044 p^{110} T^{22} + p^{132} T^{24} $
	37	$ 1 - 1287952664424930744 T^{2} + $$83\!\cdots\!16$$ T^{4} - $$36\!\cdots\!40$$ T^{6} + $$11\!\cdots\!76$$ T^{8} - $$29\!\cdots\!52$$ T^{10} + $$58\!\cdots\!94$$ T^{12} - $$29\!\cdots\!52$$ p^{22} T^{14} + $$11\!\cdots\!76$$ p^{44} T^{16} - $$36\!\cdots\!40$$ p^{66} T^{18} + $$83\!\cdots\!16$$ p^{88} T^{20} - 1287952664424930744 p^{110} T^{22} + p^{132} T^{24} $
	41	$ 1 - 2849426385631088492 T^{2} + $$48\!\cdots\!82$$ T^{4} - $$58\!\cdots\!60$$ T^{6} + $$54\!\cdots\!71$$ T^{8} - $$40\!\cdots\!28$$ T^{10} + $$24\!\cdots\!12$$ T^{12} - $$40\!\cdots\!28$$ p^{22} T^{14} + $$54\!\cdots\!71$$ p^{44} T^{16} - $$58\!\cdots\!60$$ p^{66} T^{18} + $$48\!\cdots\!82$$ p^{88} T^{20} - 2849426385631088492 p^{110} T^{22} + p^{132} T^{24} $
	43	$ ( 1 + 1147259988 T + 3438686199343149828 T^{2} + $$39\!\cdots\!68$$ T^{3} + $$63\!\cdots\!56$$ T^{4} + $$57\!\cdots\!88$$ T^{5} + $$75\!\cdots\!94$$ T^{6} + $$57\!\cdots\!88$$ p^{11} T^{7} + $$63\!\cdots\!56$$ p^{22} T^{8} + $$39\!\cdots\!68$$ p^{33} T^{9} + 3438686199343149828 p^{44} T^{10} + 1147259988 p^{55} T^{11} + p^{66} T^{12} )^{2} $
	47	$ 1 - 16240051799917962008 T^{2} + $$13\!\cdots\!84$$ T^{4} - $$79\!\cdots\!72$$ T^{6} + $$34\!\cdots\!20$$ T^{8} - $$11\!\cdots\!04$$ T^{10} + $$32\!\cdots\!10$$ T^{12} - $$11\!\cdots\!04$$ p^{22} T^{14} + $$34\!\cdots\!20$$ p^{44} T^{16} - $$79\!\cdots\!72$$ p^{66} T^{18} + $$13\!\cdots\!84$$ p^{88} T^{20} - 16240051799917962008 p^{110} T^{22} + p^{132} T^{24} $
	53	$ ( 1 + 2301031380 T + 29660125066996927962 T^{2} + $$94\!\cdots\!76$$ T^{3} + $$44\!\cdots\!47$$ T^{4} + $$16\!\cdots\!80$$ T^{5} + $$47\!\cdots\!20$$ T^{6} + $$16\!\cdots\!80$$ p^{11} T^{7} + $$44\!\cdots\!47$$ p^{22} T^{8} + $$94\!\cdots\!76$$ p^{33} T^{9} + 29660125066996927962 p^{44} T^{10} + 2301031380 p^{55} T^{11} + p^{66} T^{12} )^{2} $
	59	$ 1 - $$20\!\cdots\!00$$ T^{2} + $$18\!\cdots\!14$$ T^{4} - $$84\!\cdots\!08$$ T^{6} + $$18\!\cdots\!39$$ T^{8} + $$70\!\cdots\!32$$ T^{10} - $$13\!\cdots\!96$$ T^{12} + $$70\!\cdots\!32$$ p^{22} T^{14} + $$18\!\cdots\!39$$ p^{44} T^{16} - $$84\!\cdots\!08$$ p^{66} T^{18} + $$18\!\cdots\!14$$ p^{88} T^{20} - $$20\!\cdots\!00$$ p^{110} T^{22} + p^{132} T^{24} $
	61	$ ( 1 + 6887824972 T + $$20\!\cdots\!90$$ T^{2} + $$12\!\cdots\!00$$ T^{3} + $$18\!\cdots\!83$$ T^{4} + $$92\!\cdots\!24$$ T^{5} + $$10\!\cdots\!00$$ T^{6} + $$92\!\cdots\!24$$ p^{11} T^{7} + $$18\!\cdots\!83$$ p^{22} T^{8} + $$12\!\cdots\!00$$ p^{33} T^{9} + $$20\!\cdots\!90$$ p^{44} T^{10} + 6887824972 p^{55} T^{11} + p^{66} T^{12} )^{2} $
	67	$ 1 - $$10\!\cdots\!64$$ T^{2} + $$54\!\cdots\!62$$ T^{4} - $$17\!\cdots\!64$$ T^{6} + $$42\!\cdots\!23$$ T^{8} - $$75\!\cdots\!68$$ T^{10} + $$10\!\cdots\!28$$ T^{12} - $$75\!\cdots\!68$$ p^{22} T^{14} + $$42\!\cdots\!23$$ p^{44} T^{16} - $$17\!\cdots\!64$$ p^{66} T^{18} + $$54\!\cdots\!62$$ p^{88} T^{20} - $$10\!\cdots\!64$$ p^{110} T^{22} + p^{132} T^{24} $
	71	$ 1 - $$23\!\cdots\!80$$ T^{2} + $$25\!\cdots\!80$$ T^{4} - $$17\!\cdots\!44$$ T^{6} + $$82\!\cdots\!28$$ T^{8} - $$29\!\cdots\!16$$ T^{10} + $$77\!\cdots\!02$$ T^{12} - $$29\!\cdots\!16$$ p^{22} T^{14} + $$82\!\cdots\!28$$ p^{44} T^{16} - $$17\!\cdots\!44$$ p^{66} T^{18} + $$25\!\cdots\!80$$ p^{88} T^{20} - $$23\!\cdots\!80$$ p^{110} T^{22} + p^{132} T^{24} $
	73	$ 1 - 25272084132351987996 p T^{2} + $$16\!\cdots\!74$$ T^{4} - $$10\!\cdots\!72$$ T^{6} + $$48\!\cdots\!35$$ T^{8} - $$19\!\cdots\!24$$ T^{10} + $$67\!\cdots\!40$$ T^{12} - $$19\!\cdots\!24$$ p^{22} T^{14} + $$48\!\cdots\!35$$ p^{44} T^{16} - $$10\!\cdots\!72$$ p^{66} T^{18} + $$16\!\cdots\!74$$ p^{88} T^{20} - 25272084132351987996 p^{111} T^{22} + p^{132} T^{24} $
	79	$ ( 1 + 9023048648 T + $$32\!\cdots\!86$$ T^{2} + $$12\!\cdots\!72$$ T^{3} + $$45\!\cdots\!67$$ T^{4} + $$22\!\cdots\!24$$ T^{5} + $$40\!\cdots\!12$$ T^{6} + $$22\!\cdots\!24$$ p^{11} T^{7} + $$45\!\cdots\!67$$ p^{22} T^{8} + $$12\!\cdots\!72$$ p^{33} T^{9} + $$32\!\cdots\!86$$ p^{44} T^{10} + 9023048648 p^{55} T^{11} + p^{66} T^{12} )^{2} $
	83	$ 1 - $$75\!\cdots\!48$$ T^{2} + $$31\!\cdots\!50$$ T^{4} - $$89\!\cdots\!16$$ T^{6} + $$19\!\cdots\!51$$ T^{8} - $$33\!\cdots\!32$$ T^{10} + $$47\!\cdots\!16$$ T^{12} - $$33\!\cdots\!32$$ p^{22} T^{14} + $$19\!\cdots\!51$$ p^{44} T^{16} - $$89\!\cdots\!16$$ p^{66} T^{18} + $$31\!\cdots\!50$$ p^{88} T^{20} - $$75\!\cdots\!48$$ p^{110} T^{22} + p^{132} T^{24} $
	89	$ 1 - $$15\!\cdots\!12$$ T^{2} + $$11\!\cdots\!34$$ T^{4} - $$23\!\cdots\!64$$ T^{6} + $$17\!\cdots\!83$$ T^{8} - $$23\!\cdots\!00$$ T^{10} + $$12\!\cdots\!96$$ T^{12} - $$23\!\cdots\!00$$ p^{22} T^{14} + $$17\!\cdots\!83$$ p^{44} T^{16} - $$23\!\cdots\!64$$ p^{66} T^{18} + $$11\!\cdots\!34$$ p^{88} T^{20} - $$15\!\cdots\!12$$ p^{110} T^{22} + p^{132} T^{24} $
	97	$ 1 - $$39\!\cdots\!52$$ T^{2} + $$70\!\cdots\!74$$ T^{4} - $$72\!\cdots\!56$$ T^{6} + $$47\!\cdots\!43$$ T^{8} - $$20\!\cdots\!24$$ T^{10} + $$90\!\cdots\!64$$ T^{12} - $$20\!\cdots\!24$$ p^{22} T^{14} + $$47\!\cdots\!43$$ p^{44} T^{16} - $$72\!\cdots\!56$$ p^{66} T^{18} + $$70\!\cdots\!74$$ p^{88} T^{20} - $$39\!\cdots\!52$$ p^{110} T^{22} + p^{132} T^{24} $
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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

−2.49560702995211645017569600378, −2.39090395873555070073563448304, −2.37567929099915709413711837588, −2.26466950353022771412278108676, −2.24338567477648227669263712765, −1.95516733904526264102300638715, −1.91885361849282356644681371218, −1.76758757614183296191731692749, −1.76298471739045569459718448585, −1.60189103009303736006779546102, −1.43701221252521954307457415373, −1.24778426317118709529413927406, −1.21064996113730865426177200874, −1.19407485009768741753868861578, −1.15878962138221562208562623851, −1.08101181271596242414978509926, −0.884007212342040051325929143478, −0.790144450309655265110296881997, −0.57190935804418472820385893557, −0.55829878897004531873240587455, −0.51167911060906972741251189218, −0.44635923297323275714441791052, −0.28560215659644501384452108422, −0.18925308307208090095735925584, −0.14082572863996775430499467651, 0.14082572863996775430499467651, 0.18925308307208090095735925584, 0.28560215659644501384452108422, 0.44635923297323275714441791052, 0.51167911060906972741251189218, 0.55829878897004531873240587455, 0.57190935804418472820385893557, 0.790144450309655265110296881997, 0.884007212342040051325929143478, 1.08101181271596242414978509926, 1.15878962138221562208562623851, 1.19407485009768741753868861578, 1.21064996113730865426177200874, 1.24778426317118709529413927406, 1.43701221252521954307457415373, 1.60189103009303736006779546102, 1.76298471739045569459718448585, 1.76758757614183296191731692749, 1.91885361849282356644681371218, 1.95516733904526264102300638715, 2.24338567477648227669263712765, 2.26466950353022771412278108676, 2.37567929099915709413711837588, 2.39090395873555070073563448304, 2.49560702995211645017569600378

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 $\F_{q}$

\(L(6)\)	\(\approx\)	\(46.24772325\)
\(L(\frac12)\)	\(\approx\)	\(46.24772325\)
\(L(\frac{13}{2})\)		not available
\(L(1)\)		not available

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