LMFDB - L-function 32-300e16-1.1-c4e16-0-2

Dirichlet series

L(s) = 1

− 6·2^-s + 22·4^-s − 216·9^-s − 176·13^-s − 320·16^-s + 1.29e3·18^-s + 1.05e3·26^-s + 1.72e3·29^-s + 1.22e3·32^-s − 4.75e3·36^-s + 1.56e3·37^-s + 1.24e3·41^-s + 1.38e4·49^-s − 3.87e3·52^-s + 288·53^-s − 1.03e4·58^-s − 3.76e3·61^-s + 744·64^-s − 1.10e4·73^-s − 9.40e3·74^-s + 2.62e4·81^-s − 7.48e3·82^-s − 768·89^-s − 7.24e3·97^-s − 8.30e4·98^-s − 672·101^-s − 1.72e3·106^-s + ⋯

L(s) = 1

− 3/2·2^-s + 11/8·4^-s − 8/3·9^-s − 1.04·13^-s − 5/4·16^-s + 4·18^-s + 1.56·26^-s + 2.05·29^-s + 1.19·32^-s − 3.66·36^-s + 1.14·37^-s + 0.742·41^-s + 5.76·49^-s − 1.43·52^-s + 0.102·53^-s − 3.08·58^-s − 1.01·61^-s + 0.181·64^-s − 2.07·73^-s − 1.71·74^-s + 4·81^-s − 1.11·82^-s − 0.0969·89^-s − 0.770·97^-s − 8.65·98^-s − 0.0658·101^-s − 0.153·106^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$32$
Conductor:	$2^{32} \cdot 3^{16} \cdot 5^{32}$
Sign:	$1$
Analytic conductor:	$7.31559\times 10^{23}$
Root analytic conductor:	$5.56875$
Motivic weight:	$4$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [2]^{16} ),\ 1 )$

Particular Values

$L(\frac{5}{2})$	$\approx$	$12.96949091$
$L(\frac12)$	$\approx$	$12.96949091$
$L(3)$		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 p T + 7 p T^{2} - 3 p^{4} T^{3} - 69 p^{2} T^{4} + 3 p^{5} T^{5} + 95 p^{5} T^{6} + 63 p^{7} T^{7} - 103 p^{8} T^{8} + 63 p^{11} T^{9} + 95 p^{13} T^{10} + 3 p^{17} T^{11} - 69 p^{18} T^{12} - 3 p^{24} T^{13} + 7 p^{25} T^{14} + 3 p^{29} T^{15} + p^{32} T^{16} $
	3	$ ( 1 + p^{3} T^{2} )^{8} $
	5	$ 1 $
good	7	$ 1 - 13848 T^{2} + 98870532 T^{4} - 464441149584 T^{6} + 1581806849148970 T^{8} - 581205958644166680 p T^{10} + $$80\!\cdots\!56$$ T^{12} - $$13\!\cdots\!16$$ T^{14} + $$24\!\cdots\!59$$ T^{16} - $$13\!\cdots\!16$$ p^{8} T^{18} + $$80\!\cdots\!56$$ p^{16} T^{20} - 581205958644166680 p^{25} T^{22} + 1581806849148970 p^{32} T^{24} - 464441149584 p^{40} T^{26} + 98870532 p^{48} T^{28} - 13848 p^{56} T^{30} + p^{64} T^{32} $
	11	$ 1 - 122256 T^{2} + 7356166968 T^{4} - 288449006358960 T^{6} + 8276874217488547228 T^{8} - $$18\!\cdots\!04$$ T^{10} + $$34\!\cdots\!64$$ T^{12} - $$57\!\cdots\!44$$ T^{14} + $$86\!\cdots\!30$$ T^{16} - $$57\!\cdots\!44$$ p^{8} T^{18} + $$34\!\cdots\!64$$ p^{16} T^{20} - $$18\!\cdots\!04$$ p^{24} T^{22} + 8276874217488547228 p^{32} T^{24} - 288449006358960 p^{40} T^{26} + 7356166968 p^{48} T^{28} - 122256 p^{56} T^{30} + p^{64} T^{32} $
	13	$ ( 1 + 88 T + 142900 T^{2} + 13706480 T^{3} + 10249794554 T^{4} + 962172803464 T^{5} + 486981831795824 T^{6} + 40856272617216200 T^{7} + 16429273501104973699 T^{8} + 40856272617216200 p^{4} T^{9} + 486981831795824 p^{8} T^{10} + 962172803464 p^{12} T^{11} + 10249794554 p^{16} T^{12} + 13706480 p^{20} T^{13} + 142900 p^{24} T^{14} + 88 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	17	$ ( 1 + 416904 T^{2} + 7595520 T^{3} + 87631111036 T^{4} + 2792328783360 T^{5} + 11993258534523192 T^{6} + 433030768798725120 T^{7} + $$11\!\cdots\!38$$ T^{8} + 433030768798725120 p^{4} T^{9} + 11993258534523192 p^{8} T^{10} + 2792328783360 p^{12} T^{11} + 87631111036 p^{16} T^{12} + 7595520 p^{20} T^{13} + 416904 p^{24} T^{14} + p^{32} T^{16} )^{2} $
	19	$ 1 - 1062008 T^{2} + 532164629092 T^{4} - 169255074447940624 T^{6} + $$39\!\cdots\!30$$ T^{8} - $$73\!\cdots\!20$$ T^{10} + $$12\!\cdots\!16$$ T^{12} - $$18\!\cdots\!76$$ T^{14} + $$24\!\cdots\!19$$ T^{16} - $$18\!\cdots\!76$$ p^{8} T^{18} + $$12\!\cdots\!16$$ p^{16} T^{20} - $$73\!\cdots\!20$$ p^{24} T^{22} + $$39\!\cdots\!30$$ p^{32} T^{24} - 169255074447940624 p^{40} T^{26} + 532164629092 p^{48} T^{28} - 1062008 p^{56} T^{30} + p^{64} T^{32} $
	23	$ 1 - 1482896 T^{2} + 1349977888312 T^{4} - 851395283312295088 T^{6} + $$42\!\cdots\!68$$ T^{8} - $$17\!\cdots\!92$$ T^{10} + $$60\!\cdots\!84$$ T^{12} - $$18\!\cdots\!68$$ T^{14} + $$55\!\cdots\!54$$ T^{16} - $$18\!\cdots\!68$$ p^{8} T^{18} + $$60\!\cdots\!84$$ p^{16} T^{20} - $$17\!\cdots\!92$$ p^{24} T^{22} + $$42\!\cdots\!68$$ p^{32} T^{24} - 851395283312295088 p^{40} T^{26} + 1349977888312 p^{48} T^{28} - 1482896 p^{56} T^{30} + p^{64} T^{32} $
	29	$ ( 1 - 864 T + 3691784 T^{2} - 2471579040 T^{3} + 6536869136892 T^{4} - 3691508704448736 T^{5} + 7632540163375261624 T^{6} - $$37\!\cdots\!40$$ T^{7} + $$63\!\cdots\!66$$ T^{8} - $$37\!\cdots\!40$$ p^{4} T^{9} + 7632540163375261624 p^{8} T^{10} - 3691508704448736 p^{12} T^{11} + 6536869136892 p^{16} T^{12} - 2471579040 p^{20} T^{13} + 3691784 p^{24} T^{14} - 864 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	31	$ 1 - 8324568 T^{2} + 35406611950020 T^{4} - $$10\!\cdots\!84$$ T^{6} + $$21\!\cdots\!98$$ T^{8} - $$37\!\cdots\!24$$ T^{10} + $$53\!\cdots\!28$$ T^{12} - $$63\!\cdots\!32$$ T^{14} + $$63\!\cdots\!91$$ T^{16} - $$63\!\cdots\!32$$ p^{8} T^{18} + $$53\!\cdots\!28$$ p^{16} T^{20} - $$37\!\cdots\!24$$ p^{24} T^{22} + $$21\!\cdots\!98$$ p^{32} T^{24} - $$10\!\cdots\!84$$ p^{40} T^{26} + 35406611950020 p^{48} T^{28} - 8324568 p^{56} T^{30} + p^{64} T^{32} $
	37	$ ( 1 - 784 T + 7571896 T^{2} - 5949579056 T^{3} + 843388318700 p T^{4} - 25585441629609232 T^{5} + 88506854415521801480 T^{6} - $$70\!\cdots\!68$$ T^{7} + $$18\!\cdots\!06$$ T^{8} - $$70\!\cdots\!68$$ p^{4} T^{9} + 88506854415521801480 p^{8} T^{10} - 25585441629609232 p^{12} T^{11} + 843388318700 p^{17} T^{12} - 5949579056 p^{20} T^{13} + 7571896 p^{24} T^{14} - 784 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	41	$ ( 1 - 624 T + 229304 p T^{2} - 7366638672 T^{3} + 54561103061116 T^{4} - 38742860619832176 T^{5} + $$22\!\cdots\!96$$ T^{6} - $$15\!\cdots\!92$$ T^{7} + $$72\!\cdots\!58$$ T^{8} - $$15\!\cdots\!92$$ p^{4} T^{9} + $$22\!\cdots\!96$$ p^{8} T^{10} - 38742860619832176 p^{12} T^{11} + 54561103061116 p^{16} T^{12} - 7366638672 p^{20} T^{13} + 229304 p^{25} T^{14} - 624 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	43	$ 1 - 20941304 T^{2} + 228049937037796 T^{4} - $$17\!\cdots\!92$$ T^{6} + $$11\!\cdots\!54$$ T^{8} - $$57\!\cdots\!24$$ T^{10} + $$25\!\cdots\!36$$ T^{12} - $$10\!\cdots\!88$$ T^{14} + $$37\!\cdots\!63$$ T^{16} - $$10\!\cdots\!88$$ p^{8} T^{18} + $$25\!\cdots\!36$$ p^{16} T^{20} - $$57\!\cdots\!24$$ p^{24} T^{22} + $$11\!\cdots\!54$$ p^{32} T^{24} - $$17\!\cdots\!92$$ p^{40} T^{26} + 228049937037796 p^{48} T^{28} - 20941304 p^{56} T^{30} + p^{64} T^{32} $
	47	$ 1 - 34735664 T^{2} + 677058858479608 T^{4} - $$92\!\cdots\!12$$ T^{6} + $$98\!\cdots\!84$$ T^{8} - $$85\!\cdots\!12$$ T^{10} + $$61\!\cdots\!56$$ T^{12} - $$37\!\cdots\!36$$ T^{14} + $$19\!\cdots\!10$$ T^{16} - $$37\!\cdots\!36$$ p^{8} T^{18} + $$61\!\cdots\!56$$ p^{16} T^{20} - $$85\!\cdots\!12$$ p^{24} T^{22} + $$98\!\cdots\!84$$ p^{32} T^{24} - $$92\!\cdots\!12$$ p^{40} T^{26} + 677058858479608 p^{48} T^{28} - 34735664 p^{56} T^{30} + p^{64} T^{32} $
	53	$ ( 1 - 144 T + 37978616 T^{2} - 18114475056 T^{3} + 707198620954620 T^{4} - 457700682827611152 T^{5} + $$86\!\cdots\!60$$ T^{6} - $$59\!\cdots\!48$$ T^{7} + $$78\!\cdots\!06$$ T^{8} - $$59\!\cdots\!48$$ p^{4} T^{9} + $$86\!\cdots\!60$$ p^{8} T^{10} - 457700682827611152 p^{12} T^{11} + 707198620954620 p^{16} T^{12} - 18114475056 p^{20} T^{13} + 37978616 p^{24} T^{14} - 144 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	59	$ 1 - 132069808 T^{2} + 8553888620940792 T^{4} - $$36\!\cdots\!84$$ T^{6} + $$11\!\cdots\!60$$ T^{8} - $$45\!\cdots\!60$$ p T^{10} + $$51\!\cdots\!76$$ T^{12} - $$81\!\cdots\!16$$ T^{14} + $$10\!\cdots\!34$$ T^{16} - $$81\!\cdots\!16$$ p^{8} T^{18} + $$51\!\cdots\!76$$ p^{16} T^{20} - $$45\!\cdots\!60$$ p^{25} T^{22} + $$11\!\cdots\!60$$ p^{32} T^{24} - $$36\!\cdots\!84$$ p^{40} T^{26} + 8553888620940792 p^{48} T^{28} - 132069808 p^{56} T^{30} + p^{64} T^{32} $
	61	$ ( 1 + 1880 T + 31413940 T^{2} - 9219826448 T^{3} + 428818059848570 T^{4} - 30424869117586648 p T^{5} + $$39\!\cdots\!96$$ T^{6} - $$42\!\cdots\!40$$ T^{7} + $$37\!\cdots\!79$$ T^{8} - $$42\!\cdots\!40$$ p^{4} T^{9} + $$39\!\cdots\!96$$ p^{8} T^{10} - 30424869117586648 p^{13} T^{11} + 428818059848570 p^{16} T^{12} - 9219826448 p^{20} T^{13} + 31413940 p^{24} T^{14} + 1880 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	67	$ 1 - 1898664 p T^{2} + 9101325221324772 T^{4} - $$46\!\cdots\!64$$ T^{6} + $$18\!\cdots\!10$$ T^{8} - $$61\!\cdots\!20$$ T^{10} + $$17\!\cdots\!76$$ T^{12} - $$42\!\cdots\!36$$ T^{14} + $$91\!\cdots\!39$$ T^{16} - $$42\!\cdots\!36$$ p^{8} T^{18} + $$17\!\cdots\!76$$ p^{16} T^{20} - $$61\!\cdots\!20$$ p^{24} T^{22} + $$18\!\cdots\!10$$ p^{32} T^{24} - $$46\!\cdots\!64$$ p^{40} T^{26} + 9101325221324772 p^{48} T^{28} - 1898664 p^{57} T^{30} + p^{64} T^{32} $
	71	$ 1 - 119435568 T^{2} + 8741084066063544 T^{4} - $$47\!\cdots\!32$$ T^{6} + $$21\!\cdots\!40$$ T^{8} - $$81\!\cdots\!00$$ T^{10} + $$27\!\cdots\!08$$ T^{12} - $$82\!\cdots\!28$$ T^{14} + $$22\!\cdots\!30$$ T^{16} - $$82\!\cdots\!28$$ p^{8} T^{18} + $$27\!\cdots\!08$$ p^{16} T^{20} - $$81\!\cdots\!00$$ p^{24} T^{22} + $$21\!\cdots\!40$$ p^{32} T^{24} - $$47\!\cdots\!32$$ p^{40} T^{26} + 8741084066063544 p^{48} T^{28} - 119435568 p^{56} T^{30} + p^{64} T^{32} $
	73	$ ( 1 + 5520 T + 111454008 T^{2} + 450477762480 T^{3} + 7050260413766044 T^{4} + 26151689112051031440 T^{5} + $$31\!\cdots\!24$$ T^{6} + $$99\!\cdots\!00$$ T^{7} + $$10\!\cdots\!06$$ T^{8} + $$99\!\cdots\!00$$ p^{4} T^{9} + $$31\!\cdots\!24$$ p^{8} T^{10} + 26151689112051031440 p^{12} T^{11} + 7050260413766044 p^{16} T^{12} + 450477762480 p^{20} T^{13} + 111454008 p^{24} T^{14} + 5520 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	79	$ 1 - 258932880 T^{2} + 35098107145400184 T^{4} - $$32\!\cdots\!00$$ T^{6} + $$22\!\cdots\!16$$ T^{8} - $$12\!\cdots\!00$$ T^{10} + $$62\!\cdots\!32$$ T^{12} - $$26\!\cdots\!80$$ T^{14} + $$10\!\cdots\!58$$ T^{16} - $$26\!\cdots\!80$$ p^{8} T^{18} + $$62\!\cdots\!32$$ p^{16} T^{20} - $$12\!\cdots\!00$$ p^{24} T^{22} + $$22\!\cdots\!16$$ p^{32} T^{24} - $$32\!\cdots\!00$$ p^{40} T^{26} + 35098107145400184 p^{48} T^{28} - 258932880 p^{56} T^{30} + p^{64} T^{32} $
	83	$ 1 - 386816432 T^{2} + 78111539130746872 T^{4} - $$10\!\cdots\!92$$ T^{6} + $$11\!\cdots\!04$$ T^{8} - $$99\!\cdots\!88$$ T^{10} + $$70\!\cdots\!72$$ T^{12} - $$42\!\cdots\!20$$ T^{14} + $$21\!\cdots\!50$$ T^{16} - $$42\!\cdots\!20$$ p^{8} T^{18} + $$70\!\cdots\!72$$ p^{16} T^{20} - $$99\!\cdots\!88$$ p^{24} T^{22} + $$11\!\cdots\!04$$ p^{32} T^{24} - $$10\!\cdots\!92$$ p^{40} T^{26} + 78111539130746872 p^{48} T^{28} - 386816432 p^{56} T^{30} + p^{64} T^{32} $
	89	$ ( 1 + 384 T + 242635016 T^{2} + 537892911744 T^{3} + 31936021870136604 T^{4} + 90763846822573014912 T^{5} + $$30\!\cdots\!48$$ T^{6} + $$84\!\cdots\!92$$ T^{7} + $$21\!\cdots\!26$$ T^{8} + $$84\!\cdots\!92$$ p^{4} T^{9} + $$30\!\cdots\!48$$ p^{8} T^{10} + 90763846822573014912 p^{12} T^{11} + 31936021870136604 p^{16} T^{12} + 537892911744 p^{20} T^{13} + 242635016 p^{24} T^{14} + 384 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	97	$ ( 1 + 3624 T + 158200772 T^{2} - 219172281744 T^{3} + 28908610128992362 T^{4} + 15633481239574495128 T^{5} + $$35\!\cdots\!44$$ T^{6} - $$30\!\cdots\!24$$ T^{7} + $$32\!\cdots\!03$$ T^{8} - $$30\!\cdots\!24$$ p^{4} T^{9} + $$35\!\cdots\!44$$ p^{8} T^{10} + 15633481239574495128 p^{12} T^{11} + 28908610128992362 p^{16} T^{12} - 219172281744 p^{20} T^{13} + 158200772 p^{24} T^{14} + 3624 p^{28} T^{15} + p^{32} T^{16} )^{2} $
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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

−2.54412726320017201026562043139, −2.44372221649896400285877075280, −2.38310436582820856417236308754, −2.34941298215517742335937781811, −2.21167660667504201949076011277, −2.03194809874762860367236857431, −1.97221866144395689209771681054, −1.83751242721032340265510209523, −1.80338804128788546847913171947, −1.66025830407389607413596008806, −1.52994282597859289847129851700, −1.48735166077029710316691895794, −1.28171137127376151293806269400, −1.23856725506339496632186327214, −1.15660790643076058929157854130, −0.925085644494381122605822627978, −0.923739915938365217738454587242, −0.817387949074730910180947025335, −0.73065859758307201244181656067, −0.49000447250369396891328832436, −0.48658822595615798341007137681, −0.39869633193400481603069337737, −0.37965984876171500535436479822, −0.21357072427937922402990143113, −0.18066829528717244269768664600, 0.18066829528717244269768664600, 0.21357072427937922402990143113, 0.37965984876171500535436479822, 0.39869633193400481603069337737, 0.48658822595615798341007137681, 0.49000447250369396891328832436, 0.73065859758307201244181656067, 0.817387949074730910180947025335, 0.923739915938365217738454587242, 0.925085644494381122605822627978, 1.15660790643076058929157854130, 1.23856725506339496632186327214, 1.28171137127376151293806269400, 1.48735166077029710316691895794, 1.52994282597859289847129851700, 1.66025830407389607413596008806, 1.80338804128788546847913171947, 1.83751242721032340265510209523, 1.97221866144395689209771681054, 2.03194809874762860367236857431, 2.21167660667504201949076011277, 2.34941298215517742335937781811, 2.38310436582820856417236308754, 2.44372221649896400285877075280, 2.54412726320017201026562043139

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 $\F_{q}$

\(L(\frac{5}{2})\)	\(\approx\)	\(12.96949091\)
\(L(\frac12)\)	\(\approx\)	\(12.96949091\)
\(L(3)\)		not available
\(L(1)\)		not available

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