Properties

Label 32-300e16-1.1-c4e16-0-2
Degree $32$
Conductor $4.305\times 10^{39}$
Sign $1$
Analytic cond. $7.31559\times 10^{23}$
Root an. cond. $5.56875$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

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: induced by $\chi_{300} (1, \cdot )$
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)$
bad2 \( 1 + 3 p T + 7 p T^{2} - 3 p^{4} T^{3} - 69 p^{2} T^{4} + 3 p^{5} T^{5} + 95 p^{5} T^{6} + 63 p^{7} T^{7} - 103 p^{8} T^{8} + 63 p^{11} T^{9} + 95 p^{13} T^{10} + 3 p^{17} T^{11} - 69 p^{18} T^{12} - 3 p^{24} T^{13} + 7 p^{25} T^{14} + 3 p^{29} T^{15} + p^{32} T^{16} \)
3 \( ( 1 + p^{3} T^{2} )^{8} \)
5 \( 1 \)
good7 \( 1 - 13848 T^{2} + 98870532 T^{4} - 464441149584 T^{6} + 1581806849148970 T^{8} - 581205958644166680 p T^{10} + \)\(80\!\cdots\!56\)\( T^{12} - \)\(13\!\cdots\!16\)\( T^{14} + \)\(24\!\cdots\!59\)\( T^{16} - \)\(13\!\cdots\!16\)\( p^{8} T^{18} + \)\(80\!\cdots\!56\)\( p^{16} T^{20} - 581205958644166680 p^{25} T^{22} + 1581806849148970 p^{32} T^{24} - 464441149584 p^{40} T^{26} + 98870532 p^{48} T^{28} - 13848 p^{56} T^{30} + p^{64} T^{32} \)
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.