LMFDB - L-function 20-336e10-1.1-c5e10-0-1

Dirichlet series

L(s) = 1

− 45·3^-s − 6·5^-s + 97·7^-s + 810·9^-s − 424·11^-s + 374·13^-s + 270·15^-s − 952·17^-s − 139·19^-s − 4.36e3·21^-s − 4.28e3·23^-s + 5.02e3·25^-s − 3.64e3·27^-s − 4.21e3·29^-s − 8.13e3·31^-s + 1.90e4·33^-s − 582·35^-s − 5.42e3·37^-s − 1.68e4·39^-s + 2.93e4·41^-s + 4.68e4·43^-s − 4.86e3·45^-s + 1.71e4·47^-s + 1.63e4·49^-s + 4.28e4·51^-s + 1.50e4·53^-s + 2.54e3·55^-s + ⋯

L(s) = 1

− 2.88·3^-s − 0.107·5^-s + 0.748·7^-s + 10/3·9^-s − 1.05·11^-s + 0.613·13^-s + 0.309·15^-s − 0.798·17^-s − 0.0883·19^-s − 2.15·21^-s − 1.69·23^-s + 1.60·25^-s − 0.962·27^-s − 0.930·29^-s − 1.51·31^-s + 3.04·33^-s − 0.0803·35^-s − 0.651·37^-s − 1.77·39^-s + 2.72·41^-s + 3.86·43^-s − 0.357·45^-s + 1.13·47^-s + 0.971·49^-s + 2.30·51^-s + 0.736·53^-s + 0.113·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$20$
Conductor:	$2^{40} \cdot 3^{10} \cdot 7^{10}$
Sign:	$1$
Analytic conductor:	$2.06537\times 10^{17}$
Root analytic conductor:	$7.34091$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(20,\ 2^{40} \cdot 3^{10} \cdot 7^{10} ,\ ( \ : [5/2]^{10} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$0.3226483261$
$L(\frac12)$	$\approx$	$0.3226483261$
$L(\frac{7}{2})$		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 $
	3	$ ( 1 + p^{2} T + p^{4} T^{2} )^{5} $
	7	$ 1 - 97 T - 989 p T^{2} + 54186 p^{2} T^{3} + 740983 p^{3} T^{4} - 34846615 p^{4} T^{5} + 740983 p^{8} T^{6} + 54186 p^{12} T^{7} - 989 p^{16} T^{8} - 97 p^{20} T^{9} + p^{25} T^{10} $
good	5	$ 1 + 6 T - 4992 T^{2} - 201868 T^{3} + 2916061 p T^{4} + 915127428 T^{5} + 4594893886 p T^{6} - 2062459524878 T^{7} - 249374496430127 T^{8} + 2739004487229496 T^{9} + 1001500497592867126 T^{10} + 2739004487229496 p^{5} T^{11} - 249374496430127 p^{10} T^{12} - 2062459524878 p^{15} T^{13} + 4594893886 p^{21} T^{14} + 915127428 p^{25} T^{15} + 2916061 p^{31} T^{16} - 201868 p^{35} T^{17} - 4992 p^{40} T^{18} + 6 p^{45} T^{19} + p^{50} T^{20} $
	11	$ 1 + 424 T - 222104 T^{2} + 28491004 T^{3} + 72028206649 T^{4} - 18372729939468 T^{5} - 1179536232804714 T^{6} + 7313520298425614056 T^{7} - 61954751798660457157 p T^{8} - $$22\!\cdots\!96$$ p^{2} T^{9} + $$43\!\cdots\!54$$ T^{10} - $$22\!\cdots\!96$$ p^{7} T^{11} - 61954751798660457157 p^{11} T^{12} + 7313520298425614056 p^{15} T^{13} - 1179536232804714 p^{20} T^{14} - 18372729939468 p^{25} T^{15} + 72028206649 p^{30} T^{16} + 28491004 p^{35} T^{17} - 222104 p^{40} T^{18} + 424 p^{45} T^{19} + p^{50} T^{20} $
	13	$ ( 1 - 187 T + 1266568 T^{2} - 310622253 T^{3} + 747059650399 T^{4} - 178369633393516 T^{5} + 747059650399 p^{5} T^{6} - 310622253 p^{10} T^{7} + 1266568 p^{15} T^{8} - 187 p^{20} T^{9} + p^{25} T^{10} )^{2} $
	17	$ 1 + 56 p T - 5399541 T^{2} - 3417664424 T^{3} + 18335466557135 T^{4} + 7102664171981520 T^{5} - 45738765853767329950 T^{6} - $$96\!\cdots\!96$$ T^{7} + $$89\!\cdots\!85$$ T^{8} + $$37\!\cdots\!36$$ p T^{9} - $$14\!\cdots\!63$$ T^{10} + $$37\!\cdots\!36$$ p^{6} T^{11} + $$89\!\cdots\!85$$ p^{10} T^{12} - $$96\!\cdots\!96$$ p^{15} T^{13} - 45738765853767329950 p^{20} T^{14} + 7102664171981520 p^{25} T^{15} + 18335466557135 p^{30} T^{16} - 3417664424 p^{35} T^{17} - 5399541 p^{40} T^{18} + 56 p^{46} T^{19} + p^{50} T^{20} $
	19	$ 1 + 139 T - 7957877 T^{2} + 8106065552 T^{3} + 1920339398000 p T^{4} - 54253270104730432 T^{5} - 68763267931124027095 T^{6} + $$18\!\cdots\!99$$ T^{7} + $$25\!\cdots\!71$$ T^{8} - $$20\!\cdots\!20$$ T^{9} + $$22\!\cdots\!68$$ T^{10} - $$20\!\cdots\!20$$ p^{5} T^{11} + $$25\!\cdots\!71$$ p^{10} T^{12} + $$18\!\cdots\!99$$ p^{15} T^{13} - 68763267931124027095 p^{20} T^{14} - 54253270104730432 p^{25} T^{15} + 1920339398000 p^{31} T^{16} + 8106065552 p^{35} T^{17} - 7957877 p^{40} T^{18} + 139 p^{45} T^{19} + p^{50} T^{20} $
	23	$ 1 + 4288 T - 7048947 T^{2} - 69429297344 T^{3} - 36591675654433 T^{4} + 499582239316421376 T^{5} + $$71\!\cdots\!78$$ T^{6} - $$23\!\cdots\!68$$ T^{7} - $$60\!\cdots\!11$$ T^{8} + $$63\!\cdots\!64$$ T^{9} + $$45\!\cdots\!51$$ T^{10} + $$63\!\cdots\!64$$ p^{5} T^{11} - $$60\!\cdots\!11$$ p^{10} T^{12} - $$23\!\cdots\!68$$ p^{15} T^{13} + $$71\!\cdots\!78$$ p^{20} T^{14} + 499582239316421376 p^{25} T^{15} - 36591675654433 p^{30} T^{16} - 69429297344 p^{35} T^{17} - 7048947 p^{40} T^{18} + 4288 p^{45} T^{19} + p^{50} T^{20} $
	29	$ ( 1 + 2108 T + 28333376 T^{2} + 41960653588 T^{3} + 313036836878467 T^{4} + 1240471110459824784 T^{5} + 313036836878467 p^{5} T^{6} + 41960653588 p^{10} T^{7} + 28333376 p^{15} T^{8} + 2108 p^{20} T^{9} + p^{25} T^{10} )^{2} $
	31	$ 1 + 8131 T - 2041940 T^{2} - 217011896845 T^{3} - 1785803937817870 T^{4} - 10280285491660429639 T^{5} - $$25\!\cdots\!22$$ T^{6} + $$32\!\cdots\!27$$ T^{7} + $$14\!\cdots\!29$$ T^{8} + $$69\!\cdots\!66$$ T^{9} + $$11\!\cdots\!68$$ T^{10} + $$69\!\cdots\!66$$ p^{5} T^{11} + $$14\!\cdots\!29$$ p^{10} T^{12} + $$32\!\cdots\!27$$ p^{15} T^{13} - $$25\!\cdots\!22$$ p^{20} T^{14} - 10280285491660429639 p^{25} T^{15} - 1785803937817870 p^{30} T^{16} - 217011896845 p^{35} T^{17} - 2041940 p^{40} T^{18} + 8131 p^{45} T^{19} + p^{50} T^{20} $
	37	$ 1 + 5425 T - 256455655 T^{2} - 646582097718 T^{3} + 40683901954870034 T^{4} + 34001125788183235038 T^{5} - $$45\!\cdots\!25$$ T^{6} - $$46\!\cdots\!45$$ T^{7} + $$38\!\cdots\!19$$ T^{8} - $$41\!\cdots\!48$$ T^{9} - $$28\!\cdots\!44$$ T^{10} - $$41\!\cdots\!48$$ p^{5} T^{11} + $$38\!\cdots\!19$$ p^{10} T^{12} - $$46\!\cdots\!45$$ p^{15} T^{13} - $$45\!\cdots\!25$$ p^{20} T^{14} + 34001125788183235038 p^{25} T^{15} + 40683901954870034 p^{30} T^{16} - 646582097718 p^{35} T^{17} - 256455655 p^{40} T^{18} + 5425 p^{45} T^{19} + p^{50} T^{20} $
	41	$ ( 1 - 14682 T + 303993241 T^{2} - 4151601653536 T^{3} + 56966570505373006 T^{4} - $$54\!\cdots\!64$$ T^{5} + 56966570505373006 p^{5} T^{6} - 4151601653536 p^{10} T^{7} + 303993241 p^{15} T^{8} - 14682 p^{20} T^{9} + p^{25} T^{10} )^{2} $
	43	$ ( 1 - 23431 T + 858350394 T^{2} - 13637211146141 T^{3} + 269900127716611085 T^{4} - $$29\!\cdots\!16$$ T^{5} + 269900127716611085 p^{5} T^{6} - 13637211146141 p^{10} T^{7} + 858350394 p^{15} T^{8} - 23431 p^{20} T^{9} + p^{25} T^{10} )^{2} $
	47	$ 1 - 17190 T - 6936847 T^{2} + 9929405950750 T^{3} - 170431162359457641 T^{4} + $$95\!\cdots\!48$$ T^{5} + $$55\!\cdots\!70$$ T^{6} - $$11\!\cdots\!92$$ T^{7} + $$85\!\cdots\!49$$ T^{8} + $$19\!\cdots\!70$$ T^{9} - $$45\!\cdots\!49$$ T^{10} + $$19\!\cdots\!70$$ p^{5} T^{11} + $$85\!\cdots\!49$$ p^{10} T^{12} - $$11\!\cdots\!92$$ p^{15} T^{13} + $$55\!\cdots\!70$$ p^{20} T^{14} + $$95\!\cdots\!48$$ p^{25} T^{15} - 170431162359457641 p^{30} T^{16} + 9929405950750 p^{35} T^{17} - 6936847 p^{40} T^{18} - 17190 p^{45} T^{19} + p^{50} T^{20} $
	53	$ 1 - 15064 T - 570811800 T^{2} + 2624123712320 T^{3} + 114440820059020181 T^{4} + $$36\!\cdots\!36$$ T^{5} + $$26\!\cdots\!62$$ T^{6} - $$22\!\cdots\!00$$ T^{7} - $$87\!\cdots\!31$$ T^{8} + $$35\!\cdots\!80$$ T^{9} + $$13\!\cdots\!26$$ T^{10} + $$35\!\cdots\!80$$ p^{5} T^{11} - $$87\!\cdots\!31$$ p^{10} T^{12} - $$22\!\cdots\!00$$ p^{15} T^{13} + $$26\!\cdots\!62$$ p^{20} T^{14} + $$36\!\cdots\!36$$ p^{25} T^{15} + 114440820059020181 p^{30} T^{16} + 2624123712320 p^{35} T^{17} - 570811800 p^{40} T^{18} - 15064 p^{45} T^{19} + p^{50} T^{20} $
	59	$ 1 - 83242 T + 2595815996 T^{2} - 26050401826288 T^{3} - 983095353155961515 T^{4} + $$76\!\cdots\!96$$ T^{5} - $$30\!\cdots\!86$$ T^{6} + $$58\!\cdots\!30$$ T^{7} + $$28\!\cdots\!49$$ T^{8} - $$50\!\cdots\!96$$ T^{9} + $$17\!\cdots\!10$$ T^{10} - $$50\!\cdots\!96$$ p^{5} T^{11} + $$28\!\cdots\!49$$ p^{10} T^{12} + $$58\!\cdots\!30$$ p^{15} T^{13} - $$30\!\cdots\!86$$ p^{20} T^{14} + $$76\!\cdots\!96$$ p^{25} T^{15} - 983095353155961515 p^{30} T^{16} - 26050401826288 p^{35} T^{17} + 2595815996 p^{40} T^{18} - 83242 p^{45} T^{19} + p^{50} T^{20} $
	61	$ 1 - 14954 T - 1306557925 T^{2} + 6314194399222 T^{3} + 62506887872587647 T^{4} + $$16\!\cdots\!36$$ T^{5} + $$20\!\cdots\!18$$ T^{6} + $$18\!\cdots\!72$$ T^{7} + $$64\!\cdots\!09$$ T^{8} - $$97\!\cdots\!26$$ T^{9} - $$96\!\cdots\!39$$ T^{10} - $$97\!\cdots\!26$$ p^{5} T^{11} + $$64\!\cdots\!09$$ p^{10} T^{12} + $$18\!\cdots\!72$$ p^{15} T^{13} + $$20\!\cdots\!18$$ p^{20} T^{14} + $$16\!\cdots\!36$$ p^{25} T^{15} + 62506887872587647 p^{30} T^{16} + 6314194399222 p^{35} T^{17} - 1306557925 p^{40} T^{18} - 14954 p^{45} T^{19} + p^{50} T^{20} $
	67	$ 1 + 39501 T - 1859217173 T^{2} - 38264949635144 T^{3} + 1397422245479996772 T^{4} - $$85\!\cdots\!68$$ T^{5} - $$37\!\cdots\!75$$ T^{6} + $$19\!\cdots\!73$$ T^{7} + $$56\!\cdots\!11$$ T^{8} + $$10\!\cdots\!64$$ T^{9} - $$36\!\cdots\!80$$ T^{10} + $$10\!\cdots\!64$$ p^{5} T^{11} + $$56\!\cdots\!11$$ p^{10} T^{12} + $$19\!\cdots\!73$$ p^{15} T^{13} - $$37\!\cdots\!75$$ p^{20} T^{14} - $$85\!\cdots\!68$$ p^{25} T^{15} + 1397422245479996772 p^{30} T^{16} - 38264949635144 p^{35} T^{17} - 1859217173 p^{40} T^{18} + 39501 p^{45} T^{19} + p^{50} T^{20} $
	71	$ ( 1 - 28010 T + 5222512779 T^{2} - 131983696999592 T^{3} + 15410360709528761314 T^{4} - $$32\!\cdots\!32$$ T^{5} + 15410360709528761314 p^{5} T^{6} - 131983696999592 p^{10} T^{7} + 5222512779 p^{15} T^{8} - 28010 p^{20} T^{9} + p^{25} T^{10} )^{2} $
	73	$ 1 + 90395 T - 5312661431 T^{2} - 382945353711426 T^{3} + 46645654577732253498 T^{4} + $$19\!\cdots\!90$$ T^{5} - $$17\!\cdots\!21$$ T^{6} - $$31\!\cdots\!63$$ T^{7} + $$64\!\cdots\!67$$ T^{8} + $$52\!\cdots\!44$$ T^{9} - $$13\!\cdots\!88$$ T^{10} + $$52\!\cdots\!44$$ p^{5} T^{11} + $$64\!\cdots\!67$$ p^{10} T^{12} - $$31\!\cdots\!63$$ p^{15} T^{13} - $$17\!\cdots\!21$$ p^{20} T^{14} + $$19\!\cdots\!90$$ p^{25} T^{15} + 46645654577732253498 p^{30} T^{16} - 382945353711426 p^{35} T^{17} - 5312661431 p^{40} T^{18} + 90395 p^{45} T^{19} + p^{50} T^{20} $
	79	$ 1 - 43067 T - 1807231124 T^{2} + 178180167200229 T^{3} - 13621209791324753742 T^{4} + $$17\!\cdots\!91$$ T^{5} + $$37\!\cdots\!74$$ T^{6} - $$80\!\cdots\!55$$ T^{7} + $$44\!\cdots\!77$$ T^{8} - $$44\!\cdots\!58$$ T^{9} - $$28\!\cdots\!16$$ T^{10} - $$44\!\cdots\!58$$ p^{5} T^{11} + $$44\!\cdots\!77$$ p^{10} T^{12} - $$80\!\cdots\!55$$ p^{15} T^{13} + $$37\!\cdots\!74$$ p^{20} T^{14} + $$17\!\cdots\!91$$ p^{25} T^{15} - 13621209791324753742 p^{30} T^{16} + 178180167200229 p^{35} T^{17} - 1807231124 p^{40} T^{18} - 43067 p^{45} T^{19} + p^{50} T^{20} $
	83	$ ( 1 - 37836 T + 14976698536 T^{2} - 393374130010502 T^{3} + 99554024204046530647 T^{4} - $$19\!\cdots\!40$$ T^{5} + 99554024204046530647 p^{5} T^{6} - 393374130010502 p^{10} T^{7} + 14976698536 p^{15} T^{8} - 37836 p^{20} T^{9} + p^{25} T^{10} )^{2} $
	89	$ 1 + 72608 T - 13875478825 T^{2} - 2243092722805408 T^{3} + 70480031564695818403 T^{4} + $$26\!\cdots\!60$$ T^{5} + $$84\!\cdots\!18$$ T^{6} - $$18\!\cdots\!48$$ T^{7} - $$14\!\cdots\!55$$ T^{8} + $$44\!\cdots\!36$$ T^{9} + $$11\!\cdots\!73$$ T^{10} + $$44\!\cdots\!36$$ p^{5} T^{11} - $$14\!\cdots\!55$$ p^{10} T^{12} - $$18\!\cdots\!48$$ p^{15} T^{13} + $$84\!\cdots\!18$$ p^{20} T^{14} + $$26\!\cdots\!60$$ p^{25} T^{15} + 70480031564695818403 p^{30} T^{16} - 2243092722805408 p^{35} T^{17} - 13875478825 p^{40} T^{18} + 72608 p^{45} T^{19} + p^{50} T^{20} $
	97	$ ( 1 - 91500 T + 39317987310 T^{2} - 2836977247703574 T^{3} + $$64\!\cdots\!29$$ T^{4} - $$35\!\cdots\!84$$ T^{5} + $$64\!\cdots\!29$$ p^{5} T^{6} - 2836977247703574 p^{10} T^{7} + 39317987310 p^{15} T^{8} - 91500 p^{20} T^{9} + p^{25} T^{10} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.52267938515870214381166411833, −3.38725415621844836800697906378, −3.24846456485664692049307529787, −2.98250575870980891276024796572, −2.97000241598091434918591462942, −2.91906243856766077526836719754, −2.62948298995252092047539688386, −2.49544893575436718211094658698, −2.35525387117431397496092084010, −2.19135512409305431633079437666, −2.16634274779953633967989592300, −2.08139738944437762684531393797, −1.93564172692035738380782720774, −1.85795741382620708447515067772, −1.54395443229671068155392854587, −1.17907657034727866547869966177, −1.12604923028314657449960771338, −1.06551999861815974487941751322, −1.05546475114740820952051868262, −0.65927872248629539262868625578, −0.60161858683948577168052177448, −0.57767608504449574929572480517, −0.52441766937056080810310644395, −0.23134949975945789309820884796, −0.05013316636526785009384671227, 0.05013316636526785009384671227, 0.23134949975945789309820884796, 0.52441766937056080810310644395, 0.57767608504449574929572480517, 0.60161858683948577168052177448, 0.65927872248629539262868625578, 1.05546475114740820952051868262, 1.06551999861815974487941751322, 1.12604923028314657449960771338, 1.17907657034727866547869966177, 1.54395443229671068155392854587, 1.85795741382620708447515067772, 1.93564172692035738380782720774, 2.08139738944437762684531393797, 2.16634274779953633967989592300, 2.19135512409305431633079437666, 2.35525387117431397496092084010, 2.49544893575436718211094658698, 2.62948298995252092047539688386, 2.91906243856766077526836719754, 2.97000241598091434918591462942, 2.98250575870980891276024796572, 3.24846456485664692049307529787, 3.38725415621844836800697906378, 3.52267938515870214381166411833

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.3226483261\)
\(L(\frac12)\)	\(\approx\)	\(0.3226483261\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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