LMFDB - L-function 32-765e16-1.1-c3e16-0-0

Dirichlet series

L(s) = 1

− 2·2^-s − 13·4^-s + 8·8^-s + 48·13^-s + 51·16^-s + 132·17^-s − 44·19^-s − 200·25^-s − 96·26^-s + 658·32^-s − 264·34^-s + 88·38^-s − 92·43^-s − 268·47^-s + 2.24e3·49^-s + 400·50^-s − 624·52^-s + 328·53^-s − 1.06e3·59^-s − 1.61e3·64^-s − 284·67^-s − 1.71e3·68^-s + 572·76^-s − 3.66e3·83^-s + 184·86^-s + 2.46e3·89^-s + 536·94^-s + ⋯

L(s) = 1

− 0.707·2^-s − 1.62·4^-s + 0.353·8^-s + 1.02·13^-s + 0.796·16^-s + 1.88·17^-s − 0.531·19^-s − 8/5·25^-s − 0.724·26^-s + 3.63·32^-s − 1.33·34^-s + 0.375·38^-s − 0.326·43^-s − 0.831·47^-s + 6.53·49^-s + 1.13·50^-s − 1.66·52^-s + 0.850·53^-s − 2.33·59^-s − 3.14·64^-s − 0.517·67^-s − 3.06·68^-s + 0.863·76^-s − 4.85·83^-s + 0.230·86^-s + 2.93·89^-s + 0.588·94^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$32$
Conductor:	$3^{32} \cdot 5^{16} \cdot 17^{16}$
Sign:	$1$
Analytic conductor:	$2.96783\times 10^{26}$
Root analytic conductor:	$6.71836$
Motivic weight:	$3$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(32,\ 3^{32} \cdot 5^{16} \cdot 17^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )$

Particular Values

$L(2)$	$\approx$	$1.811984401$
$L(\frac12)$	$\approx$	$1.811984401$
$L(\frac{5}{2})$		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 $
	5	$ ( 1 + p^{2} T^{2} )^{8} $
	17	$ 1 - 132 T + 12692 T^{2} - 1150428 T^{3} + 4397588 p T^{4} - 20126868 p^{2} T^{5} + 78795692 p^{3} T^{6} - 257274636 p^{4} T^{7} + 1107246582 p^{5} T^{8} - 257274636 p^{7} T^{9} + 78795692 p^{9} T^{10} - 20126868 p^{11} T^{11} + 4397588 p^{13} T^{12} - 1150428 p^{15} T^{13} + 12692 p^{18} T^{14} - 132 p^{21} T^{15} + p^{24} T^{16} $
good	2	$ ( 1 + T + p^{3} T^{2} + 9 p T^{3} + 79 T^{4} - 75 T^{5} + 95 p^{3} T^{6} + 15 p^{3} T^{7} + 115 p^{5} T^{8} + 15 p^{6} T^{9} + 95 p^{9} T^{10} - 75 p^{9} T^{11} + 79 p^{12} T^{12} + 9 p^{16} T^{13} + p^{21} T^{14} + p^{21} T^{15} + p^{24} T^{16} )^{2} $
	7	$ 1 - 320 p T^{2} + 2771396 T^{4} - 2421626008 T^{6} + 1659777731524 T^{8} - 941322775552200 T^{10} + 455489391805595180 T^{12} - 27296223011222085344 p T^{14} + $$70\!\cdots\!38$$ T^{16} - 27296223011222085344 p^{7} T^{18} + 455489391805595180 p^{12} T^{20} - 941322775552200 p^{18} T^{22} + 1659777731524 p^{24} T^{24} - 2421626008 p^{30} T^{26} + 2771396 p^{36} T^{28} - 320 p^{43} T^{30} + p^{48} T^{32} $
	11	$ 1 - 8532 T^{2} + 41831028 T^{4} - 147710972676 T^{6} + 409184217237108 T^{8} - 933220315120529676 T^{10} + $$17\!\cdots\!08$$ T^{12} - $$24\!\cdots\!44$$ p^{2} T^{14} + $$42\!\cdots\!46$$ T^{16} - $$24\!\cdots\!44$$ p^{8} T^{18} + $$17\!\cdots\!08$$ p^{12} T^{20} - 933220315120529676 p^{18} T^{22} + 409184217237108 p^{24} T^{24} - 147710972676 p^{30} T^{26} + 41831028 p^{36} T^{28} - 8532 p^{42} T^{30} + p^{48} T^{32} $
	13	$ ( 1 - 24 T + 7615 T^{2} - 151228 T^{3} + 30025482 T^{4} - 747085980 T^{5} + 6800773917 p T^{6} - 2591897158416 T^{7} + 212219862441466 T^{8} - 2591897158416 p^{3} T^{9} + 6800773917 p^{7} T^{10} - 747085980 p^{9} T^{11} + 30025482 p^{12} T^{12} - 151228 p^{15} T^{13} + 7615 p^{18} T^{14} - 24 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	19	$ ( 1 + 22 T + 14485 T^{2} - 14034 T^{3} + 176856190 T^{4} + 2988581870 T^{5} + 1805466966275 T^{6} + 11986414346790 T^{7} + 12300936829448738 T^{8} + 11986414346790 p^{3} T^{9} + 1805466966275 p^{6} T^{10} + 2988581870 p^{9} T^{11} + 176856190 p^{12} T^{12} - 14034 p^{15} T^{13} + 14485 p^{18} T^{14} + 22 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	23	$ 1 - 127184 T^{2} + 7999103380 T^{4} - 330756641524824 T^{6} + 10075708965129045572 T^{8} - $$23\!\cdots\!44$$ T^{10} + $$46\!\cdots\!16$$ T^{12} - $$73\!\cdots\!12$$ T^{14} + $$97\!\cdots\!22$$ T^{16} - $$73\!\cdots\!12$$ p^{6} T^{18} + $$46\!\cdots\!16$$ p^{12} T^{20} - $$23\!\cdots\!44$$ p^{18} T^{22} + 10075708965129045572 p^{24} T^{24} - 330756641524824 p^{30} T^{26} + 7999103380 p^{36} T^{28} - 127184 p^{42} T^{30} + p^{48} T^{32} $
	29	$ 1 - 79078 T^{2} + 2955331573 T^{4} - 57560277167166 T^{6} + 13668416513355582 p T^{8} + $$90\!\cdots\!62$$ T^{10} + $$46\!\cdots\!67$$ T^{12} - $$32\!\cdots\!94$$ T^{14} + $$10\!\cdots\!54$$ T^{16} - $$32\!\cdots\!94$$ p^{6} T^{18} + $$46\!\cdots\!67$$ p^{12} T^{20} + $$90\!\cdots\!62$$ p^{18} T^{22} + 13668416513355582 p^{25} T^{24} - 57560277167166 p^{30} T^{26} + 2955331573 p^{36} T^{28} - 79078 p^{42} T^{30} + p^{48} T^{32} $
	31	$ 1 - 322578 T^{2} + 50375743433 T^{4} - 5071125582168862 T^{6} + $$37\!\cdots\!34$$ T^{8} - $$20\!\cdots\!18$$ T^{10} + $$95\!\cdots\!59$$ T^{12} - $$36\!\cdots\!42$$ T^{14} + $$11\!\cdots\!06$$ T^{16} - $$36\!\cdots\!42$$ p^{6} T^{18} + $$95\!\cdots\!59$$ p^{12} T^{20} - $$20\!\cdots\!18$$ p^{18} T^{22} + $$37\!\cdots\!34$$ p^{24} T^{24} - 5071125582168862 p^{30} T^{26} + 50375743433 p^{36} T^{28} - 322578 p^{42} T^{30} + p^{48} T^{32} $
	37	$ 1 - 336408 T^{2} + 56065649016 T^{4} - 5810219985923080 T^{6} + $$38\!\cdots\!28$$ T^{8} - $$13\!\cdots\!44$$ T^{10} - $$27\!\cdots\!28$$ T^{12} + $$70\!\cdots\!00$$ T^{14} - $$48\!\cdots\!34$$ T^{16} + $$70\!\cdots\!00$$ p^{6} T^{18} - $$27\!\cdots\!28$$ p^{12} T^{20} - $$13\!\cdots\!44$$ p^{18} T^{22} + $$38\!\cdots\!28$$ p^{24} T^{24} - 5810219985923080 p^{30} T^{26} + 56065649016 p^{36} T^{28} - 336408 p^{42} T^{30} + p^{48} T^{32} $
	41	$ 1 - 545304 T^{2} + 148850236504 T^{4} - 27335059794438600 T^{6} + $$38\!\cdots\!00$$ T^{8} - $$43\!\cdots\!00$$ T^{10} + $$42\!\cdots\!00$$ T^{12} - $$35\!\cdots\!16$$ T^{14} + $$25\!\cdots\!10$$ T^{16} - $$35\!\cdots\!16$$ p^{6} T^{18} + $$42\!\cdots\!00$$ p^{12} T^{20} - $$43\!\cdots\!00$$ p^{18} T^{22} + $$38\!\cdots\!00$$ p^{24} T^{24} - 27335059794438600 p^{30} T^{26} + 148850236504 p^{36} T^{28} - 545304 p^{42} T^{30} + p^{48} T^{32} $
	43	$ ( 1 + 46 T + 245140 T^{2} - 169718 p T^{3} + 37522401724 T^{4} - 863572474306 T^{5} + 4444473974843484 T^{6} - 123512965923958386 T^{7} + $$38\!\cdots\!50$$ T^{8} - 123512965923958386 p^{3} T^{9} + 4444473974843484 p^{6} T^{10} - 863572474306 p^{9} T^{11} + 37522401724 p^{12} T^{12} - 169718 p^{16} T^{13} + 245140 p^{18} T^{14} + 46 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	47	$ ( 1 + 134 T + 371915 T^{2} + 29220264 T^{3} + 78630152230 T^{4} + 4603196433996 T^{5} + 12315246336092221 T^{6} + 639139837682351706 T^{7} + $$14\!\cdots\!78$$ T^{8} + 639139837682351706 p^{3} T^{9} + 12315246336092221 p^{6} T^{10} + 4603196433996 p^{9} T^{11} + 78630152230 p^{12} T^{12} + 29220264 p^{15} T^{13} + 371915 p^{18} T^{14} + 134 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	53	$ ( 1 - 164 T + 962895 T^{2} - 128218676 T^{3} + 420861254826 T^{4} - 45692778600860 T^{5} + 111403870677765961 T^{6} - 9987187975589474380 T^{7} + $$19\!\cdots\!90$$ T^{8} - 9987187975589474380 p^{3} T^{9} + 111403870677765961 p^{6} T^{10} - 45692778600860 p^{9} T^{11} + 420861254826 p^{12} T^{12} - 128218676 p^{15} T^{13} + 962895 p^{18} T^{14} - 164 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	59	$ ( 1 + 530 T + 1198109 T^{2} + 553661194 T^{3} + 676377293582 T^{4} + 268391300522650 T^{5} + 237940012112777131 T^{6} + 80535343575657706866 T^{7} + $$57\!\cdots\!54$$ T^{8} + 80535343575657706866 p^{3} T^{9} + 237940012112777131 p^{6} T^{10} + 268391300522650 p^{9} T^{11} + 676377293582 p^{12} T^{12} + 553661194 p^{15} T^{13} + 1198109 p^{18} T^{14} + 530 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	61	$ 1 - 2252254 T^{2} + 2579557276325 T^{4} - 1973373956079689758 T^{6} + $$11\!\cdots\!78$$ T^{8} - $$50\!\cdots\!10$$ T^{10} + $$18\!\cdots\!79$$ T^{12} - $$53\!\cdots\!58$$ T^{14} + $$13\!\cdots\!74$$ T^{16} - $$53\!\cdots\!58$$ p^{6} T^{18} + $$18\!\cdots\!79$$ p^{12} T^{20} - $$50\!\cdots\!10$$ p^{18} T^{22} + $$11\!\cdots\!78$$ p^{24} T^{24} - 1973373956079689758 p^{30} T^{26} + 2579557276325 p^{36} T^{28} - 2252254 p^{42} T^{30} + p^{48} T^{32} $
	67	$ ( 1 + 142 T + 1299880 T^{2} + 212811126 T^{3} + 890902810244 T^{4} + 148075846027718 T^{5} + 423655297633299080 T^{6} + 63891395920440807598 T^{7} + $$14\!\cdots\!06$$ T^{8} + 63891395920440807598 p^{3} T^{9} + 423655297633299080 p^{6} T^{10} + 148075846027718 p^{9} T^{11} + 890902810244 p^{12} T^{12} + 212811126 p^{15} T^{13} + 1299880 p^{18} T^{14} + 142 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	71	$ 1 - 2503514 T^{2} + 3157814189497 T^{4} - 2730320440947833758 T^{6} + $$18\!\cdots\!38$$ T^{8} - $$10\!\cdots\!34$$ T^{10} + $$50\!\cdots\!71$$ T^{12} - $$21\!\cdots\!02$$ T^{14} + $$82\!\cdots\!62$$ T^{16} - $$21\!\cdots\!02$$ p^{6} T^{18} + $$50\!\cdots\!71$$ p^{12} T^{20} - $$10\!\cdots\!34$$ p^{18} T^{22} + $$18\!\cdots\!38$$ p^{24} T^{24} - 2730320440947833758 p^{30} T^{26} + 3157814189497 p^{36} T^{28} - 2503514 p^{42} T^{30} + p^{48} T^{32} $
	73	$ 1 - 3198662 T^{2} + 4984848965069 T^{4} - 5072710998580416198 T^{6} + $$38\!\cdots\!22$$ T^{8} - $$22\!\cdots\!58$$ T^{10} + $$11\!\cdots\!71$$ T^{12} - $$48\!\cdots\!86$$ T^{14} + $$19\!\cdots\!54$$ T^{16} - $$48\!\cdots\!86$$ p^{6} T^{18} + $$11\!\cdots\!71$$ p^{12} T^{20} - $$22\!\cdots\!58$$ p^{18} T^{22} + $$38\!\cdots\!22$$ p^{24} T^{24} - 5072710998580416198 p^{30} T^{26} + 4984848965069 p^{36} T^{28} - 3198662 p^{42} T^{30} + p^{48} T^{32} $
	79	$ 1 - 3438708 T^{2} + 6127930845476 T^{4} - 7406837703912403828 T^{6} + $$67\!\cdots\!60$$ T^{8} - $$49\!\cdots\!96$$ T^{10} + $$31\!\cdots\!04$$ T^{12} - $$17\!\cdots\!04$$ T^{14} + $$88\!\cdots\!10$$ T^{16} - $$17\!\cdots\!04$$ p^{6} T^{18} + $$31\!\cdots\!04$$ p^{12} T^{20} - $$49\!\cdots\!96$$ p^{18} T^{22} + $$67\!\cdots\!60$$ p^{24} T^{24} - 7406837703912403828 p^{30} T^{26} + 6127930845476 p^{36} T^{28} - 3438708 p^{42} T^{30} + p^{48} T^{32} $
	83	$ ( 1 + 1834 T + 3988416 T^{2} + 4802316890 T^{3} + 6061745945620 T^{4} + 5457522277317818 T^{5} + 5213730935030966000 T^{6} + $$39\!\cdots\!30$$ T^{7} + $$32\!\cdots\!10$$ T^{8} + $$39\!\cdots\!30$$ p^{3} T^{9} + 5213730935030966000 p^{6} T^{10} + 5457522277317818 p^{9} T^{11} + 6061745945620 p^{12} T^{12} + 4802316890 p^{15} T^{13} + 3988416 p^{18} T^{14} + 1834 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	89	$ ( 1 - 1234 T + 4545537 T^{2} - 4310254310 T^{3} + 9220148464810 T^{4} - 7050750976675274 T^{5} + 11300963744326660127 T^{6} - $$71\!\cdots\!14$$ T^{7} + $$94\!\cdots\!10$$ T^{8} - $$71\!\cdots\!14$$ p^{3} T^{9} + 11300963744326660127 p^{6} T^{10} - 7050750976675274 p^{9} T^{11} + 9220148464810 p^{12} T^{12} - 4310254310 p^{15} T^{13} + 4545537 p^{18} T^{14} - 1234 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	97	$ 1 - 9728078 T^{2} + 46166581065949 T^{4} - $$14\!\cdots\!18$$ T^{6} + $$32\!\cdots\!42$$ T^{8} - $$57\!\cdots\!78$$ T^{10} + $$83\!\cdots\!87$$ T^{12} - $$99\!\cdots\!58$$ T^{14} + $$10\!\cdots\!98$$ p^{2} T^{16} - $$99\!\cdots\!58$$ p^{6} T^{18} + $$83\!\cdots\!87$$ p^{12} T^{20} - $$57\!\cdots\!78$$ p^{18} T^{22} + $$32\!\cdots\!42$$ p^{24} T^{24} - $$14\!\cdots\!18$$ p^{30} T^{26} + 46166581065949 p^{36} T^{28} - 9728078 p^{42} T^{30} + p^{48} T^{32} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−2.13371709035285805180274833626, −2.08873535584384812780388109883, −2.07746982515022080352803053576, −2.03940171205569929777761341116, −1.94463962344115687480492579233, −1.88035992671655027203060385066, −1.71816108183753902258870741873, −1.67061380541791293816021738346, −1.64653229936137048489445809597, −1.51536440237787289380874128355, −1.29358660121728800562250332776, −1.23314020742339361594120357998, −1.09824244084888899998693534948, −1.08389031044144355386377209816, −0.976179188453511609114485724929, −0.857846708139213536461132440172, −0.848166056077320320589155923999, −0.69212770096575331968992425873, −0.64613720214773705055432438175, −0.61859936107288227660233976688, −0.39560351663420226722124733047, −0.38886392541006725427809889402, −0.35075507411518984712850086597, −0.19176176909321400772971806659, −0.05042123475051111155081330751, 0.05042123475051111155081330751, 0.19176176909321400772971806659, 0.35075507411518984712850086597, 0.38886392541006725427809889402, 0.39560351663420226722124733047, 0.61859936107288227660233976688, 0.64613720214773705055432438175, 0.69212770096575331968992425873, 0.848166056077320320589155923999, 0.857846708139213536461132440172, 0.976179188453511609114485724929, 1.08389031044144355386377209816, 1.09824244084888899998693534948, 1.23314020742339361594120357998, 1.29358660121728800562250332776, 1.51536440237787289380874128355, 1.64653229936137048489445809597, 1.67061380541791293816021738346, 1.71816108183753902258870741873, 1.88035992671655027203060385066, 1.94463962344115687480492579233, 2.03940171205569929777761341116, 2.07746982515022080352803053576, 2.08873535584384812780388109883, 2.13371709035285805180274833626

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.811984401\)
\(L(\frac12)\)	\(\approx\)	\(1.811984401\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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