Properties

Label 16-63e8-1.1-c6e8-0-2
Degree $16$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $1.94699\times 10^{9}$
Root an. cond. $3.80702$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 54·4-s + 294·5-s − 656·7-s − 667·8-s + 1.47e3·10-s + 314·11-s − 3.28e3·14-s − 3.48e3·16-s + 5.53e3·17-s − 1.82e4·19-s + 1.58e4·20-s + 1.57e3·22-s − 3.92e3·23-s + 3.44e3·25-s − 3.54e4·28-s + 8.30e3·29-s − 8.95e4·31-s − 2.92e4·32-s + 2.76e4·34-s − 1.92e5·35-s + 6.47e4·37-s − 9.11e4·38-s − 1.96e5·40-s + 4.57e4·43-s + 1.69e4·44-s − 1.96e4·46-s + ⋯
L(s)  = 1  + 5/8·2-s + 0.843·4-s + 2.35·5-s − 1.91·7-s − 1.30·8-s + 1.46·10-s + 0.235·11-s − 1.19·14-s − 0.850·16-s + 1.12·17-s − 2.65·19-s + 1.98·20-s + 0.147·22-s − 0.322·23-s + 0.220·25-s − 1.61·28-s + 0.340·29-s − 3.00·31-s − 0.892·32-s + 0.703·34-s − 4.49·35-s + 1.27·37-s − 1.66·38-s − 3.06·40-s + 0.575·43-s + 0.199·44-s − 0.201·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.94699\times 10^{9}\)
Root analytic conductor: \(3.80702\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [3]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(7.249785286\)
\(L(\frac12)\) \(\approx\) \(7.249785286\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + 656 T + 52930 p T^{2} + 309592 p^{3} T^{3} + 2489441 p^{5} T^{4} + 309592 p^{9} T^{5} + 52930 p^{13} T^{6} + 656 p^{18} T^{7} + p^{24} T^{8} \)
good2 \( 1 - 5 T - 29 T^{2} + 541 p T^{3} - 231 p^{4} T^{4} - 5931 p^{3} T^{5} + 31781 p^{4} T^{6} + 555 p^{7} T^{7} - 4201 p^{13} T^{8} + 555 p^{13} T^{9} + 31781 p^{16} T^{10} - 5931 p^{21} T^{11} - 231 p^{28} T^{12} + 541 p^{31} T^{13} - 29 p^{36} T^{14} - 5 p^{42} T^{15} + p^{48} T^{16} \)
5 \( 1 - 294 T + 82987 T^{2} - 637098 p^{2} T^{3} + 117579973 p^{2} T^{4} - 774031572 p^{4} T^{5} + 117124815442 p^{4} T^{6} - 3317043950544 p^{5} T^{7} + 84831264587506 p^{6} T^{8} - 3317043950544 p^{11} T^{9} + 117124815442 p^{16} T^{10} - 774031572 p^{22} T^{11} + 117579973 p^{26} T^{12} - 637098 p^{32} T^{13} + 82987 p^{36} T^{14} - 294 p^{42} T^{15} + p^{48} T^{16} \)
11 \( 1 - 314 T - 3241301 T^{2} - 2282591446 T^{3} + 5834825775945 T^{4} + 6616089629332092 T^{5} + 3272307340128160526 T^{6} - \)\(89\!\cdots\!28\)\( T^{7} - \)\(14\!\cdots\!74\)\( T^{8} - \)\(89\!\cdots\!28\)\( p^{6} T^{9} + 3272307340128160526 p^{12} T^{10} + 6616089629332092 p^{18} T^{11} + 5834825775945 p^{24} T^{12} - 2282591446 p^{30} T^{13} - 3241301 p^{36} T^{14} - 314 p^{42} T^{15} + p^{48} T^{16} \)
13 \( 1 - 8061410 T^{2} + 99428085070849 T^{4} - 42524425799434012010 p T^{6} + \)\(35\!\cdots\!16\)\( T^{8} - 42524425799434012010 p^{13} T^{10} + 99428085070849 p^{24} T^{12} - 8061410 p^{36} T^{14} + p^{48} T^{16} \)
17 \( 1 - 5532 T + 88822624 T^{2} - 434934779712 T^{3} + 4293664204523122 T^{4} - 21974649003770301900 T^{5} + \)\(15\!\cdots\!92\)\( T^{6} - \)\(73\!\cdots\!80\)\( T^{7} + \)\(40\!\cdots\!75\)\( T^{8} - \)\(73\!\cdots\!80\)\( p^{6} T^{9} + \)\(15\!\cdots\!92\)\( p^{12} T^{10} - 21974649003770301900 p^{18} T^{11} + 4293664204523122 p^{24} T^{12} - 434934779712 p^{30} T^{13} + 88822624 p^{36} T^{14} - 5532 p^{42} T^{15} + p^{48} T^{16} \)
19 \( 1 + 18234 T + 243472639 T^{2} + 2418674220558 T^{3} + 18439013730415501 T^{4} + 96957691664786483076 T^{5} + \)\(31\!\cdots\!02\)\( T^{6} - \)\(41\!\cdots\!56\)\( T^{7} - \)\(10\!\cdots\!26\)\( T^{8} - \)\(41\!\cdots\!56\)\( p^{6} T^{9} + \)\(31\!\cdots\!02\)\( p^{12} T^{10} + 96957691664786483076 p^{18} T^{11} + 18439013730415501 p^{24} T^{12} + 2418674220558 p^{30} T^{13} + 243472639 p^{36} T^{14} + 18234 p^{42} T^{15} + p^{48} T^{16} \)
23 \( 1 + 3928 T - 430822724 T^{2} - 1912742625712 T^{3} + 103644170278515594 T^{4} + \)\(39\!\cdots\!00\)\( T^{5} - \)\(17\!\cdots\!24\)\( T^{6} - \)\(29\!\cdots\!08\)\( T^{7} + \)\(25\!\cdots\!27\)\( T^{8} - \)\(29\!\cdots\!08\)\( p^{6} T^{9} - \)\(17\!\cdots\!24\)\( p^{12} T^{10} + \)\(39\!\cdots\!00\)\( p^{18} T^{11} + 103644170278515594 p^{24} T^{12} - 1912742625712 p^{30} T^{13} - 430822724 p^{36} T^{14} + 3928 p^{42} T^{15} + p^{48} T^{16} \)
29 \( ( 1 - 4150 T + 1032394617 T^{2} + 18866311211278 T^{3} + 388636276445225152 T^{4} + 18866311211278 p^{6} T^{5} + 1032394617 p^{12} T^{6} - 4150 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
31 \( 1 + 89508 T + 6988306546 T^{2} + 386472796257864 T^{3} + 19782003758288014117 T^{4} + \)\(82\!\cdots\!96\)\( T^{5} + \)\(32\!\cdots\!74\)\( T^{6} + \)\(11\!\cdots\!68\)\( T^{7} + \)\(35\!\cdots\!92\)\( T^{8} + \)\(11\!\cdots\!68\)\( p^{6} T^{9} + \)\(32\!\cdots\!74\)\( p^{12} T^{10} + \)\(82\!\cdots\!96\)\( p^{18} T^{11} + 19782003758288014117 p^{24} T^{12} + 386472796257864 p^{30} T^{13} + 6988306546 p^{36} T^{14} + 89508 p^{42} T^{15} + p^{48} T^{16} \)
37 \( 1 - 64706 T - 6529328325 T^{2} + 234246855700578 T^{3} + 43484194599829556697 T^{4} - \)\(81\!\cdots\!36\)\( T^{5} - \)\(16\!\cdots\!34\)\( T^{6} + \)\(67\!\cdots\!12\)\( T^{7} + \)\(49\!\cdots\!34\)\( T^{8} + \)\(67\!\cdots\!12\)\( p^{6} T^{9} - \)\(16\!\cdots\!34\)\( p^{12} T^{10} - \)\(81\!\cdots\!36\)\( p^{18} T^{11} + 43484194599829556697 p^{24} T^{12} + 234246855700578 p^{30} T^{13} - 6529328325 p^{36} T^{14} - 64706 p^{42} T^{15} + p^{48} T^{16} \)
41 \( 1 - 16407804632 T^{2} + \)\(17\!\cdots\!84\)\( T^{4} - \)\(12\!\cdots\!84\)\( T^{6} + \)\(66\!\cdots\!46\)\( T^{8} - \)\(12\!\cdots\!84\)\( p^{12} T^{10} + \)\(17\!\cdots\!84\)\( p^{24} T^{12} - 16407804632 p^{36} T^{14} + p^{48} T^{16} \)
43 \( ( 1 - 22870 T + 20444241385 T^{2} - 348583585734614 T^{3} + \)\(17\!\cdots\!32\)\( T^{4} - 348583585734614 p^{6} T^{5} + 20444241385 p^{12} T^{6} - 22870 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
47 \( 1 + 483276 T + 134436866848 T^{2} + 27346157698817856 T^{3} + \)\(43\!\cdots\!50\)\( T^{4} + \)\(59\!\cdots\!64\)\( T^{5} + \)\(70\!\cdots\!20\)\( T^{6} + \)\(77\!\cdots\!00\)\( T^{7} + \)\(80\!\cdots\!99\)\( T^{8} + \)\(77\!\cdots\!00\)\( p^{6} T^{9} + \)\(70\!\cdots\!20\)\( p^{12} T^{10} + \)\(59\!\cdots\!64\)\( p^{18} T^{11} + \)\(43\!\cdots\!50\)\( p^{24} T^{12} + 27346157698817856 p^{30} T^{13} + 134436866848 p^{36} T^{14} + 483276 p^{42} T^{15} + p^{48} T^{16} \)
53 \( 1 - 540974 T + 115320896035 T^{2} - 16822557538791898 T^{3} + \)\(32\!\cdots\!45\)\( T^{4} - \)\(54\!\cdots\!68\)\( T^{5} + \)\(60\!\cdots\!94\)\( T^{6} - \)\(10\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!46\)\( T^{8} - \)\(10\!\cdots\!48\)\( p^{6} T^{9} + \)\(60\!\cdots\!94\)\( p^{12} T^{10} - \)\(54\!\cdots\!68\)\( p^{18} T^{11} + \)\(32\!\cdots\!45\)\( p^{24} T^{12} - 16822557538791898 p^{30} T^{13} + 115320896035 p^{36} T^{14} - 540974 p^{42} T^{15} + p^{48} T^{16} \)
59 \( 1 - 181770 T + 105759390787 T^{2} - 17221970692941990 T^{3} + \)\(55\!\cdots\!17\)\( T^{4} - \)\(12\!\cdots\!24\)\( T^{5} + \)\(26\!\cdots\!02\)\( T^{6} - \)\(75\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!86\)\( T^{8} - \)\(75\!\cdots\!68\)\( p^{6} T^{9} + \)\(26\!\cdots\!02\)\( p^{12} T^{10} - \)\(12\!\cdots\!24\)\( p^{18} T^{11} + \)\(55\!\cdots\!17\)\( p^{24} T^{12} - 17221970692941990 p^{30} T^{13} + 105759390787 p^{36} T^{14} - 181770 p^{42} T^{15} + p^{48} T^{16} \)
61 \( 1 - 418224 T + 94454636356 T^{2} - 15119159348703936 T^{3} - \)\(15\!\cdots\!34\)\( T^{4} + \)\(41\!\cdots\!24\)\( T^{5} + \)\(10\!\cdots\!28\)\( T^{6} - \)\(83\!\cdots\!72\)\( T^{7} + \)\(30\!\cdots\!63\)\( T^{8} - \)\(83\!\cdots\!72\)\( p^{6} T^{9} + \)\(10\!\cdots\!28\)\( p^{12} T^{10} + \)\(41\!\cdots\!24\)\( p^{18} T^{11} - \)\(15\!\cdots\!34\)\( p^{24} T^{12} - 15119159348703936 p^{30} T^{13} + 94454636356 p^{36} T^{14} - 418224 p^{42} T^{15} + p^{48} T^{16} \)
67 \( 1 + 1158902 T + 546211729431 T^{2} + 178210845253494490 T^{3} + \)\(73\!\cdots\!69\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{5} + \)\(96\!\cdots\!34\)\( T^{6} + \)\(29\!\cdots\!64\)\( T^{7} + \)\(94\!\cdots\!02\)\( T^{8} + \)\(29\!\cdots\!64\)\( p^{6} T^{9} + \)\(96\!\cdots\!34\)\( p^{12} T^{10} + \)\(30\!\cdots\!60\)\( p^{18} T^{11} + \)\(73\!\cdots\!69\)\( p^{24} T^{12} + 178210845253494490 p^{30} T^{13} + 546211729431 p^{36} T^{14} + 1158902 p^{42} T^{15} + p^{48} T^{16} \)
71 \( ( 1 + 721172 T + 475842934968 T^{2} + 161860951437779356 T^{3} + \)\(69\!\cdots\!62\)\( T^{4} + 161860951437779356 p^{6} T^{5} + 475842934968 p^{12} T^{6} + 721172 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
73 \( 1 + 378666 T + 420023261659 T^{2} + 140949815892729462 T^{3} + \)\(82\!\cdots\!29\)\( T^{4} + \)\(30\!\cdots\!04\)\( T^{5} + \)\(43\!\cdots\!62\)\( T^{6} - \)\(37\!\cdots\!48\)\( T^{7} - \)\(64\!\cdots\!58\)\( T^{8} - \)\(37\!\cdots\!48\)\( p^{6} T^{9} + \)\(43\!\cdots\!62\)\( p^{12} T^{10} + \)\(30\!\cdots\!04\)\( p^{18} T^{11} + \)\(82\!\cdots\!29\)\( p^{24} T^{12} + 140949815892729462 p^{30} T^{13} + 420023261659 p^{36} T^{14} + 378666 p^{42} T^{15} + p^{48} T^{16} \)
79 \( 1 - 611452 T - 451137876462 T^{2} + 193948375837473256 T^{3} + \)\(18\!\cdots\!29\)\( T^{4} - \)\(28\!\cdots\!72\)\( T^{5} - \)\(65\!\cdots\!74\)\( T^{6} + \)\(38\!\cdots\!92\)\( p T^{7} + \)\(27\!\cdots\!36\)\( p^{2} T^{8} + \)\(38\!\cdots\!92\)\( p^{7} T^{9} - \)\(65\!\cdots\!74\)\( p^{12} T^{10} - \)\(28\!\cdots\!72\)\( p^{18} T^{11} + \)\(18\!\cdots\!29\)\( p^{24} T^{12} + 193948375837473256 p^{30} T^{13} - 451137876462 p^{36} T^{14} - 611452 p^{42} T^{15} + p^{48} T^{16} \)
83 \( 1 - 2090775792938 T^{2} + \)\(19\!\cdots\!41\)\( T^{4} - \)\(11\!\cdots\!82\)\( T^{6} + \)\(43\!\cdots\!80\)\( T^{8} - \)\(11\!\cdots\!82\)\( p^{12} T^{10} + \)\(19\!\cdots\!41\)\( p^{24} T^{12} - 2090775792938 p^{36} T^{14} + p^{48} T^{16} \)
89 \( 1 - 989196 T + 2419085186752 T^{2} - 2070303751015730880 T^{3} + \)\(32\!\cdots\!42\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{5} + \)\(27\!\cdots\!32\)\( T^{6} - \)\(17\!\cdots\!56\)\( T^{7} + \)\(16\!\cdots\!39\)\( T^{8} - \)\(17\!\cdots\!56\)\( p^{6} T^{9} + \)\(27\!\cdots\!32\)\( p^{12} T^{10} - \)\(23\!\cdots\!92\)\( p^{18} T^{11} + \)\(32\!\cdots\!42\)\( p^{24} T^{12} - 2070303751015730880 p^{30} T^{13} + 2419085186752 p^{36} T^{14} - 989196 p^{42} T^{15} + p^{48} T^{16} \)
97 \( 1 - 4607066646866 T^{2} + \)\(10\!\cdots\!37\)\( T^{4} - \)\(14\!\cdots\!54\)\( T^{6} + \)\(14\!\cdots\!60\)\( T^{8} - \)\(14\!\cdots\!54\)\( p^{12} T^{10} + \)\(10\!\cdots\!37\)\( p^{24} T^{12} - 4607066646866 p^{36} T^{14} + p^{48} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.86131044555329052050656446237, −5.66078693311674411328784975524, −5.60626902981548116075054527826, −5.50721049272415562628027025024, −4.91686105344580029544038256685, −4.81480658813910743693385265995, −4.55709613354967179436399463833, −4.49272122013782799553849069733, −4.10505951126567602353563704921, −3.86007482302288796605811666622, −3.54929716766722112290819308130, −3.44136491673692982782234934955, −3.20818868486663485054658514777, −3.20540510577608200615066048016, −2.94803213154187876750503103359, −2.42242259667701309379621733962, −2.24545076830203798062738217743, −2.21515898145143059941719260838, −1.98449752174967151408048713275, −1.68750319953531449292381052926, −1.60241484255832131452342435639, −1.11126042508092688136442943578, −0.44968019693561803627288935383, −0.37658757978952929228405418191, −0.36114864621370906664187662365, 0.36114864621370906664187662365, 0.37658757978952929228405418191, 0.44968019693561803627288935383, 1.11126042508092688136442943578, 1.60241484255832131452342435639, 1.68750319953531449292381052926, 1.98449752174967151408048713275, 2.21515898145143059941719260838, 2.24545076830203798062738217743, 2.42242259667701309379621733962, 2.94803213154187876750503103359, 3.20540510577608200615066048016, 3.20818868486663485054658514777, 3.44136491673692982782234934955, 3.54929716766722112290819308130, 3.86007482302288796605811666622, 4.10505951126567602353563704921, 4.49272122013782799553849069733, 4.55709613354967179436399463833, 4.81480658813910743693385265995, 4.91686105344580029544038256685, 5.50721049272415562628027025024, 5.60626902981548116075054527826, 5.66078693311674411328784975524, 5.86131044555329052050656446237

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.