Properties

Label 32-1617e16-1.1-c3e16-0-2
Degree $32$
Conductor $2.185\times 10^{51}$
Sign $1$
Analytic cond. $4.71213\times 10^{31}$
Root an. cond. $9.76760$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $16$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 48·3-s − 20·4-s − 40·5-s − 192·6-s − 90·8-s + 1.22e3·9-s − 160·10-s + 176·11-s + 960·12-s − 104·13-s + 1.92e3·15-s + 135·16-s − 180·17-s + 4.89e3·18-s − 152·19-s + 800·20-s + 704·22-s + 4·23-s + 4.32e3·24-s + 94·25-s − 416·26-s − 2.20e4·27-s + 412·29-s + 7.68e3·30-s − 628·31-s + 708·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 9.23·3-s − 5/2·4-s − 3.57·5-s − 13.0·6-s − 3.97·8-s + 45.3·9-s − 5.05·10-s + 4.82·11-s + 23.0·12-s − 2.21·13-s + 33.0·15-s + 2.10·16-s − 2.56·17-s + 64.1·18-s − 1.83·19-s + 8.94·20-s + 6.82·22-s + 0.0362·23-s + 36.7·24-s + 0.751·25-s − 3.13·26-s − 157.·27-s + 2.63·29-s + 46.7·30-s − 3.63·31-s + 3.91·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 7^{32} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(4.71213\times 10^{31}\)
Root analytic conductor: \(9.76760\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(16\)
Selberg data: \((32,\ 3^{16} \cdot 7^{32} \cdot 11^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p T )^{16} \)
7 \( 1 \)
11 \( ( 1 - p T )^{16} \)
good2 \( 1 - p^{2} T + 9 p^{2} T^{2} - 67 p T^{3} + 761 T^{4} - 663 p^{2} T^{5} + 11997 T^{6} - 19643 p T^{7} + 78699 p T^{8} - 121701 p^{2} T^{9} + 1783439 T^{10} - 327623 p^{4} T^{11} + 1122389 p^{4} T^{12} - 1570281 p^{5} T^{13} + 10214477 p^{4} T^{14} - 853231 p^{9} T^{15} + 5325245 p^{8} T^{16} - 853231 p^{12} T^{17} + 10214477 p^{10} T^{18} - 1570281 p^{14} T^{19} + 1122389 p^{16} T^{20} - 327623 p^{19} T^{21} + 1783439 p^{18} T^{22} - 121701 p^{23} T^{23} + 78699 p^{25} T^{24} - 19643 p^{28} T^{25} + 11997 p^{30} T^{26} - 663 p^{35} T^{27} + 761 p^{36} T^{28} - 67 p^{40} T^{29} + 9 p^{44} T^{30} - p^{47} T^{31} + p^{48} T^{32} \)
5 \( 1 + 8 p T + 1506 T^{2} + 40108 T^{3} + 198522 p T^{4} + 20758872 T^{5} + 409277034 T^{6} + 7248970152 T^{7} + 122325311177 T^{8} + 1907513437252 T^{9} + 28563727485808 T^{10} + 401389638566252 T^{11} + 5443835695315814 T^{12} + 13977776900911504 p T^{13} + 173740728130868532 p T^{14} + 2053249282975640876 p T^{15} + \)\(11\!\cdots\!48\)\( T^{16} + 2053249282975640876 p^{4} T^{17} + 173740728130868532 p^{7} T^{18} + 13977776900911504 p^{10} T^{19} + 5443835695315814 p^{12} T^{20} + 401389638566252 p^{15} T^{21} + 28563727485808 p^{18} T^{22} + 1907513437252 p^{21} T^{23} + 122325311177 p^{24} T^{24} + 7248970152 p^{27} T^{25} + 409277034 p^{30} T^{26} + 20758872 p^{33} T^{27} + 198522 p^{37} T^{28} + 40108 p^{39} T^{29} + 1506 p^{42} T^{30} + 8 p^{46} T^{31} + p^{48} T^{32} \)
13 \( 1 + 8 p T + 25652 T^{2} + 1943184 T^{3} + 279130654 T^{4} + 16124021432 T^{5} + 1781810145360 T^{6} + 79381828257816 T^{7} + 7795017371133297 T^{8} + 267294898729036520 T^{9} + 26384331011517156904 T^{10} + \)\(70\!\cdots\!20\)\( T^{11} + \)\(59\!\cdots\!42\)\( p T^{12} + \)\(17\!\cdots\!76\)\( T^{13} + \)\(20\!\cdots\!12\)\( T^{14} + \)\(40\!\cdots\!52\)\( T^{15} + \)\(47\!\cdots\!04\)\( T^{16} + \)\(40\!\cdots\!52\)\( p^{3} T^{17} + \)\(20\!\cdots\!12\)\( p^{6} T^{18} + \)\(17\!\cdots\!76\)\( p^{9} T^{19} + \)\(59\!\cdots\!42\)\( p^{13} T^{20} + \)\(70\!\cdots\!20\)\( p^{15} T^{21} + 26384331011517156904 p^{18} T^{22} + 267294898729036520 p^{21} T^{23} + 7795017371133297 p^{24} T^{24} + 79381828257816 p^{27} T^{25} + 1781810145360 p^{30} T^{26} + 16124021432 p^{33} T^{27} + 279130654 p^{36} T^{28} + 1943184 p^{39} T^{29} + 25652 p^{42} T^{30} + 8 p^{46} T^{31} + p^{48} T^{32} \)
17 \( 1 + 180 T + 54198 T^{2} + 7957236 T^{3} + 1451362768 T^{4} + 181850061036 T^{5} + 25448994074426 T^{6} + 2795276612168940 T^{7} + 326768347236760204 T^{8} + 32004380790033019140 T^{9} + \)\(19\!\cdots\!02\)\( p T^{10} + \)\(28\!\cdots\!08\)\( T^{11} + \)\(26\!\cdots\!88\)\( T^{12} + \)\(20\!\cdots\!12\)\( T^{13} + \)\(16\!\cdots\!66\)\( T^{14} + \)\(12\!\cdots\!00\)\( T^{15} + \)\(53\!\cdots\!62\)\( p T^{16} + \)\(12\!\cdots\!00\)\( p^{3} T^{17} + \)\(16\!\cdots\!66\)\( p^{6} T^{18} + \)\(20\!\cdots\!12\)\( p^{9} T^{19} + \)\(26\!\cdots\!88\)\( p^{12} T^{20} + \)\(28\!\cdots\!08\)\( p^{15} T^{21} + \)\(19\!\cdots\!02\)\( p^{19} T^{22} + 32004380790033019140 p^{21} T^{23} + 326768347236760204 p^{24} T^{24} + 2795276612168940 p^{27} T^{25} + 25448994074426 p^{30} T^{26} + 181850061036 p^{33} T^{27} + 1451362768 p^{36} T^{28} + 7957236 p^{39} T^{29} + 54198 p^{42} T^{30} + 180 p^{45} T^{31} + p^{48} T^{32} \)
19 \( 1 + 8 p T + 72362 T^{2} + 7737068 T^{3} + 2200706030 T^{4} + 164945433760 T^{5} + 38804955600054 T^{6} + 1798778642354344 T^{7} + 453186557872646361 T^{8} + 6241087305607852980 T^{9} + \)\(37\!\cdots\!04\)\( T^{10} - \)\(12\!\cdots\!84\)\( T^{11} + \)\(22\!\cdots\!74\)\( T^{12} - \)\(25\!\cdots\!24\)\( T^{13} + \)\(10\!\cdots\!40\)\( T^{14} - \)\(25\!\cdots\!88\)\( T^{15} + \)\(51\!\cdots\!72\)\( T^{16} - \)\(25\!\cdots\!88\)\( p^{3} T^{17} + \)\(10\!\cdots\!40\)\( p^{6} T^{18} - \)\(25\!\cdots\!24\)\( p^{9} T^{19} + \)\(22\!\cdots\!74\)\( p^{12} T^{20} - \)\(12\!\cdots\!84\)\( p^{15} T^{21} + \)\(37\!\cdots\!04\)\( p^{18} T^{22} + 6241087305607852980 p^{21} T^{23} + 453186557872646361 p^{24} T^{24} + 1798778642354344 p^{27} T^{25} + 38804955600054 p^{30} T^{26} + 164945433760 p^{33} T^{27} + 2200706030 p^{36} T^{28} + 7737068 p^{39} T^{29} + 72362 p^{42} T^{30} + 8 p^{46} T^{31} + p^{48} T^{32} \)
23 \( 1 - 4 T + 92096 T^{2} + 851484 T^{3} + 181757852 p T^{4} + 96146813964 T^{5} + 126657388879536 T^{6} + 198373717336820 p T^{7} + 2933207092863655876 T^{8} + \)\(13\!\cdots\!08\)\( T^{9} + \)\(56\!\cdots\!68\)\( T^{10} + \)\(13\!\cdots\!20\)\( p T^{11} + \)\(92\!\cdots\!92\)\( T^{12} + \)\(55\!\cdots\!64\)\( T^{13} + \)\(13\!\cdots\!88\)\( T^{14} + \)\(81\!\cdots\!04\)\( T^{15} + \)\(17\!\cdots\!26\)\( T^{16} + \)\(81\!\cdots\!04\)\( p^{3} T^{17} + \)\(13\!\cdots\!88\)\( p^{6} T^{18} + \)\(55\!\cdots\!64\)\( p^{9} T^{19} + \)\(92\!\cdots\!92\)\( p^{12} T^{20} + \)\(13\!\cdots\!20\)\( p^{16} T^{21} + \)\(56\!\cdots\!68\)\( p^{18} T^{22} + \)\(13\!\cdots\!08\)\( p^{21} T^{23} + 2933207092863655876 p^{24} T^{24} + 198373717336820 p^{28} T^{25} + 126657388879536 p^{30} T^{26} + 96146813964 p^{33} T^{27} + 181757852 p^{37} T^{28} + 851484 p^{39} T^{29} + 92096 p^{42} T^{30} - 4 p^{45} T^{31} + p^{48} T^{32} \)
29 \( 1 - 412 T + 300056 T^{2} - 86571584 T^{3} + 36823296078 T^{4} - 8095039495788 T^{5} + 2611046195789656 T^{6} - 454346770907438348 T^{7} + \)\(12\!\cdots\!97\)\( T^{8} - \)\(17\!\cdots\!72\)\( T^{9} + \)\(46\!\cdots\!80\)\( T^{10} - \)\(54\!\cdots\!48\)\( T^{11} + \)\(14\!\cdots\!06\)\( T^{12} - \)\(15\!\cdots\!56\)\( T^{13} + \)\(41\!\cdots\!20\)\( T^{14} - \)\(39\!\cdots\!88\)\( T^{15} + \)\(10\!\cdots\!68\)\( T^{16} - \)\(39\!\cdots\!88\)\( p^{3} T^{17} + \)\(41\!\cdots\!20\)\( p^{6} T^{18} - \)\(15\!\cdots\!56\)\( p^{9} T^{19} + \)\(14\!\cdots\!06\)\( p^{12} T^{20} - \)\(54\!\cdots\!48\)\( p^{15} T^{21} + \)\(46\!\cdots\!80\)\( p^{18} T^{22} - \)\(17\!\cdots\!72\)\( p^{21} T^{23} + \)\(12\!\cdots\!97\)\( p^{24} T^{24} - 454346770907438348 p^{27} T^{25} + 2611046195789656 p^{30} T^{26} - 8095039495788 p^{33} T^{27} + 36823296078 p^{36} T^{28} - 86571584 p^{39} T^{29} + 300056 p^{42} T^{30} - 412 p^{45} T^{31} + p^{48} T^{32} \)
31 \( 1 + 628 T + 443802 T^{2} + 178346516 T^{3} + 74374585860 T^{4} + 22592241795284 T^{5} + 7086480563914142 T^{6} + 1769074867387298804 T^{7} + \)\(46\!\cdots\!96\)\( T^{8} + \)\(99\!\cdots\!40\)\( T^{9} + \)\(22\!\cdots\!34\)\( T^{10} + \)\(43\!\cdots\!96\)\( T^{11} + \)\(90\!\cdots\!56\)\( T^{12} + \)\(16\!\cdots\!68\)\( T^{13} + \)\(31\!\cdots\!06\)\( T^{14} + \)\(52\!\cdots\!96\)\( T^{15} + \)\(96\!\cdots\!06\)\( T^{16} + \)\(52\!\cdots\!96\)\( p^{3} T^{17} + \)\(31\!\cdots\!06\)\( p^{6} T^{18} + \)\(16\!\cdots\!68\)\( p^{9} T^{19} + \)\(90\!\cdots\!56\)\( p^{12} T^{20} + \)\(43\!\cdots\!96\)\( p^{15} T^{21} + \)\(22\!\cdots\!34\)\( p^{18} T^{22} + \)\(99\!\cdots\!40\)\( p^{21} T^{23} + \)\(46\!\cdots\!96\)\( p^{24} T^{24} + 1769074867387298804 p^{27} T^{25} + 7086480563914142 p^{30} T^{26} + 22592241795284 p^{33} T^{27} + 74374585860 p^{36} T^{28} + 178346516 p^{39} T^{29} + 443802 p^{42} T^{30} + 628 p^{45} T^{31} + p^{48} T^{32} \)
37 \( 1 - 4 p T + 411080 T^{2} - 70256444 T^{3} + 83559740790 T^{4} - 16074136192876 T^{5} + 11346747767975180 T^{6} - 64294568607886452 p T^{7} + \)\(11\!\cdots\!21\)\( T^{8} - \)\(25\!\cdots\!32\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} - \)\(21\!\cdots\!64\)\( T^{11} + \)\(72\!\cdots\!74\)\( T^{12} - \)\(15\!\cdots\!48\)\( T^{13} + \)\(45\!\cdots\!16\)\( T^{14} - \)\(90\!\cdots\!40\)\( T^{15} + \)\(25\!\cdots\!64\)\( T^{16} - \)\(90\!\cdots\!40\)\( p^{3} T^{17} + \)\(45\!\cdots\!16\)\( p^{6} T^{18} - \)\(15\!\cdots\!48\)\( p^{9} T^{19} + \)\(72\!\cdots\!74\)\( p^{12} T^{20} - \)\(21\!\cdots\!64\)\( p^{15} T^{21} + \)\(10\!\cdots\!00\)\( p^{18} T^{22} - \)\(25\!\cdots\!32\)\( p^{21} T^{23} + \)\(11\!\cdots\!21\)\( p^{24} T^{24} - 64294568607886452 p^{28} T^{25} + 11346747767975180 p^{30} T^{26} - 16074136192876 p^{33} T^{27} + 83559740790 p^{36} T^{28} - 70256444 p^{39} T^{29} + 411080 p^{42} T^{30} - 4 p^{46} T^{31} + p^{48} T^{32} \)
41 \( 1 + 596 T + 558130 T^{2} + 284481652 T^{3} + 160254133820 T^{4} + 68353164664380 T^{5} + 29609167020868838 T^{6} + 10810190321839250780 T^{7} + \)\(39\!\cdots\!84\)\( T^{8} + \)\(12\!\cdots\!76\)\( T^{9} + \)\(39\!\cdots\!78\)\( T^{10} + \)\(11\!\cdots\!56\)\( T^{11} + \)\(77\!\cdots\!16\)\( p T^{12} + \)\(84\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!86\)\( T^{14} + \)\(57\!\cdots\!12\)\( T^{15} + \)\(15\!\cdots\!34\)\( T^{16} + \)\(57\!\cdots\!12\)\( p^{3} T^{17} + \)\(22\!\cdots\!86\)\( p^{6} T^{18} + \)\(84\!\cdots\!00\)\( p^{9} T^{19} + \)\(77\!\cdots\!16\)\( p^{13} T^{20} + \)\(11\!\cdots\!56\)\( p^{15} T^{21} + \)\(39\!\cdots\!78\)\( p^{18} T^{22} + \)\(12\!\cdots\!76\)\( p^{21} T^{23} + \)\(39\!\cdots\!84\)\( p^{24} T^{24} + 10810190321839250780 p^{27} T^{25} + 29609167020868838 p^{30} T^{26} + 68353164664380 p^{33} T^{27} + 160254133820 p^{36} T^{28} + 284481652 p^{39} T^{29} + 558130 p^{42} T^{30} + 596 p^{45} T^{31} + p^{48} T^{32} \)
43 \( 1 + 260 T + 364520 T^{2} + 55410228 T^{3} + 52760767684 T^{4} + 3409970749748 T^{5} + 4384742136277704 T^{6} - 32727427717038300 T^{7} + \)\(26\!\cdots\!40\)\( T^{8} - \)\(12\!\cdots\!56\)\( T^{9} + \)\(13\!\cdots\!76\)\( T^{10} + \)\(81\!\cdots\!28\)\( T^{11} + \)\(95\!\cdots\!20\)\( T^{12} + \)\(36\!\cdots\!36\)\( T^{13} + \)\(88\!\cdots\!48\)\( T^{14} + \)\(46\!\cdots\!40\)\( T^{15} + \)\(73\!\cdots\!86\)\( T^{16} + \)\(46\!\cdots\!40\)\( p^{3} T^{17} + \)\(88\!\cdots\!48\)\( p^{6} T^{18} + \)\(36\!\cdots\!36\)\( p^{9} T^{19} + \)\(95\!\cdots\!20\)\( p^{12} T^{20} + \)\(81\!\cdots\!28\)\( p^{15} T^{21} + \)\(13\!\cdots\!76\)\( p^{18} T^{22} - \)\(12\!\cdots\!56\)\( p^{21} T^{23} + \)\(26\!\cdots\!40\)\( p^{24} T^{24} - 32727427717038300 p^{27} T^{25} + 4384742136277704 p^{30} T^{26} + 3409970749748 p^{33} T^{27} + 52760767684 p^{36} T^{28} + 55410228 p^{39} T^{29} + 364520 p^{42} T^{30} + 260 p^{45} T^{31} + p^{48} T^{32} \)
47 \( 1 + 2220 T + 3250782 T^{2} + 3488867780 T^{3} + 3068601127966 T^{4} + 2279830005108700 T^{5} + 1480141931179634422 T^{6} + \)\(85\!\cdots\!12\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} + \)\(20\!\cdots\!60\)\( T^{9} + \)\(89\!\cdots\!80\)\( T^{10} + \)\(35\!\cdots\!08\)\( T^{11} + \)\(13\!\cdots\!54\)\( T^{12} + \)\(47\!\cdots\!56\)\( T^{13} + \)\(16\!\cdots\!48\)\( T^{14} + \)\(52\!\cdots\!28\)\( T^{15} + \)\(17\!\cdots\!12\)\( T^{16} + \)\(52\!\cdots\!28\)\( p^{3} T^{17} + \)\(16\!\cdots\!48\)\( p^{6} T^{18} + \)\(47\!\cdots\!56\)\( p^{9} T^{19} + \)\(13\!\cdots\!54\)\( p^{12} T^{20} + \)\(35\!\cdots\!08\)\( p^{15} T^{21} + \)\(89\!\cdots\!80\)\( p^{18} T^{22} + \)\(20\!\cdots\!60\)\( p^{21} T^{23} + \)\(44\!\cdots\!21\)\( p^{24} T^{24} + \)\(85\!\cdots\!12\)\( p^{27} T^{25} + 1480141931179634422 p^{30} T^{26} + 2279830005108700 p^{33} T^{27} + 3068601127966 p^{36} T^{28} + 3488867780 p^{39} T^{29} + 3250782 p^{42} T^{30} + 2220 p^{45} T^{31} + p^{48} T^{32} \)
53 \( 1 - 168 T + 1584380 T^{2} - 377434888 T^{3} + 1226513279676 T^{4} - 366057928565944 T^{5} + 623474858770816212 T^{6} - \)\(21\!\cdots\!24\)\( T^{7} + \)\(23\!\cdots\!76\)\( T^{8} - \)\(85\!\cdots\!96\)\( T^{9} + \)\(69\!\cdots\!72\)\( T^{10} - \)\(25\!\cdots\!40\)\( T^{11} + \)\(16\!\cdots\!44\)\( T^{12} - \)\(58\!\cdots\!04\)\( T^{13} + \)\(33\!\cdots\!36\)\( T^{14} - \)\(10\!\cdots\!24\)\( T^{15} + \)\(54\!\cdots\!46\)\( T^{16} - \)\(10\!\cdots\!24\)\( p^{3} T^{17} + \)\(33\!\cdots\!36\)\( p^{6} T^{18} - \)\(58\!\cdots\!04\)\( p^{9} T^{19} + \)\(16\!\cdots\!44\)\( p^{12} T^{20} - \)\(25\!\cdots\!40\)\( p^{15} T^{21} + \)\(69\!\cdots\!72\)\( p^{18} T^{22} - \)\(85\!\cdots\!96\)\( p^{21} T^{23} + \)\(23\!\cdots\!76\)\( p^{24} T^{24} - \)\(21\!\cdots\!24\)\( p^{27} T^{25} + 623474858770816212 p^{30} T^{26} - 366057928565944 p^{33} T^{27} + 1226513279676 p^{36} T^{28} - 377434888 p^{39} T^{29} + 1584380 p^{42} T^{30} - 168 p^{45} T^{31} + p^{48} T^{32} \)
59 \( 1 - 48 T + 1866728 T^{2} - 25998216 T^{3} + 1736544062022 T^{4} + 39208048973064 T^{5} + 1073950381186450240 T^{6} + 61873064288651078904 T^{7} + \)\(49\!\cdots\!61\)\( T^{8} + \)\(43\!\cdots\!72\)\( T^{9} + \)\(18\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!80\)\( T^{11} + \)\(55\!\cdots\!22\)\( T^{12} + \)\(11\!\cdots\!24\)\( p T^{13} + \)\(14\!\cdots\!48\)\( T^{14} + \)\(30\!\cdots\!68\)\( p T^{15} + \)\(31\!\cdots\!80\)\( T^{16} + \)\(30\!\cdots\!68\)\( p^{4} T^{17} + \)\(14\!\cdots\!48\)\( p^{6} T^{18} + \)\(11\!\cdots\!24\)\( p^{10} T^{19} + \)\(55\!\cdots\!22\)\( p^{12} T^{20} + \)\(20\!\cdots\!80\)\( p^{15} T^{21} + \)\(18\!\cdots\!52\)\( p^{18} T^{22} + \)\(43\!\cdots\!72\)\( p^{21} T^{23} + \)\(49\!\cdots\!61\)\( p^{24} T^{24} + 61873064288651078904 p^{27} T^{25} + 1073950381186450240 p^{30} T^{26} + 39208048973064 p^{33} T^{27} + 1736544062022 p^{36} T^{28} - 25998216 p^{39} T^{29} + 1866728 p^{42} T^{30} - 48 p^{45} T^{31} + p^{48} T^{32} \)
61 \( 1 + 1504 T + 3647736 T^{2} + 4149308096 T^{3} + 5835114087232 T^{4} + 5401405020722688 T^{5} + 5646444277738491752 T^{6} + \)\(44\!\cdots\!44\)\( T^{7} + \)\(37\!\cdots\!68\)\( T^{8} + \)\(25\!\cdots\!32\)\( T^{9} + \)\(18\!\cdots\!04\)\( T^{10} + \)\(11\!\cdots\!12\)\( T^{11} + \)\(73\!\cdots\!48\)\( T^{12} + \)\(39\!\cdots\!16\)\( T^{13} + \)\(22\!\cdots\!48\)\( T^{14} + \)\(11\!\cdots\!76\)\( T^{15} + \)\(57\!\cdots\!94\)\( T^{16} + \)\(11\!\cdots\!76\)\( p^{3} T^{17} + \)\(22\!\cdots\!48\)\( p^{6} T^{18} + \)\(39\!\cdots\!16\)\( p^{9} T^{19} + \)\(73\!\cdots\!48\)\( p^{12} T^{20} + \)\(11\!\cdots\!12\)\( p^{15} T^{21} + \)\(18\!\cdots\!04\)\( p^{18} T^{22} + \)\(25\!\cdots\!32\)\( p^{21} T^{23} + \)\(37\!\cdots\!68\)\( p^{24} T^{24} + \)\(44\!\cdots\!44\)\( p^{27} T^{25} + 5646444277738491752 p^{30} T^{26} + 5401405020722688 p^{33} T^{27} + 5835114087232 p^{36} T^{28} + 4149308096 p^{39} T^{29} + 3647736 p^{42} T^{30} + 1504 p^{45} T^{31} + p^{48} T^{32} \)
67 \( 1 - 116 T + 3164788 T^{2} - 225157040 T^{3} + 4954414589886 T^{4} - 176099421633332 T^{5} + 5104447486609003152 T^{6} - 42785692623306968076 T^{7} + \)\(38\!\cdots\!57\)\( T^{8} + \)\(64\!\cdots\!16\)\( p T^{9} + \)\(23\!\cdots\!04\)\( T^{10} + \)\(55\!\cdots\!72\)\( T^{11} + \)\(11\!\cdots\!58\)\( T^{12} + \)\(34\!\cdots\!96\)\( T^{13} + \)\(43\!\cdots\!72\)\( T^{14} + \)\(14\!\cdots\!72\)\( T^{15} + \)\(14\!\cdots\!76\)\( T^{16} + \)\(14\!\cdots\!72\)\( p^{3} T^{17} + \)\(43\!\cdots\!72\)\( p^{6} T^{18} + \)\(34\!\cdots\!96\)\( p^{9} T^{19} + \)\(11\!\cdots\!58\)\( p^{12} T^{20} + \)\(55\!\cdots\!72\)\( p^{15} T^{21} + \)\(23\!\cdots\!04\)\( p^{18} T^{22} + \)\(64\!\cdots\!16\)\( p^{22} T^{23} + \)\(38\!\cdots\!57\)\( p^{24} T^{24} - 42785692623306968076 p^{27} T^{25} + 5104447486609003152 p^{30} T^{26} - 176099421633332 p^{33} T^{27} + 4954414589886 p^{36} T^{28} - 225157040 p^{39} T^{29} + 3164788 p^{42} T^{30} - 116 p^{45} T^{31} + p^{48} T^{32} \)
71 \( 1 - 320 T + 3943664 T^{2} - 918758384 T^{3} + 7346862897712 T^{4} - 1090792261595952 T^{5} + 8644471900115051440 T^{6} - \)\(56\!\cdots\!80\)\( T^{7} + \)\(72\!\cdots\!56\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{9} + \)\(46\!\cdots\!72\)\( T^{10} + \)\(41\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!12\)\( T^{12} + \)\(35\!\cdots\!92\)\( T^{13} + \)\(10\!\cdots\!16\)\( T^{14} + \)\(18\!\cdots\!12\)\( T^{15} + \)\(40\!\cdots\!46\)\( T^{16} + \)\(18\!\cdots\!12\)\( p^{3} T^{17} + \)\(10\!\cdots\!16\)\( p^{6} T^{18} + \)\(35\!\cdots\!92\)\( p^{9} T^{19} + \)\(24\!\cdots\!12\)\( p^{12} T^{20} + \)\(41\!\cdots\!52\)\( p^{15} T^{21} + \)\(46\!\cdots\!72\)\( p^{18} T^{22} + \)\(10\!\cdots\!48\)\( p^{21} T^{23} + \)\(72\!\cdots\!56\)\( p^{24} T^{24} - \)\(56\!\cdots\!80\)\( p^{27} T^{25} + 8644471900115051440 p^{30} T^{26} - 1090792261595952 p^{33} T^{27} + 7346862897712 p^{36} T^{28} - 918758384 p^{39} T^{29} + 3943664 p^{42} T^{30} - 320 p^{45} T^{31} + p^{48} T^{32} \)
73 \( 1 + 652 T + 3493702 T^{2} + 2440833764 T^{3} + 6129207383826 T^{4} + 4485717947654876 T^{5} + 7259804293940170810 T^{6} + \)\(53\!\cdots\!40\)\( T^{7} + \)\(65\!\cdots\!33\)\( T^{8} + \)\(47\!\cdots\!64\)\( T^{9} + \)\(47\!\cdots\!04\)\( T^{10} + \)\(44\!\cdots\!20\)\( p T^{11} + \)\(28\!\cdots\!54\)\( T^{12} + \)\(18\!\cdots\!72\)\( T^{13} + \)\(14\!\cdots\!52\)\( T^{14} + \)\(84\!\cdots\!00\)\( T^{15} + \)\(60\!\cdots\!36\)\( T^{16} + \)\(84\!\cdots\!00\)\( p^{3} T^{17} + \)\(14\!\cdots\!52\)\( p^{6} T^{18} + \)\(18\!\cdots\!72\)\( p^{9} T^{19} + \)\(28\!\cdots\!54\)\( p^{12} T^{20} + \)\(44\!\cdots\!20\)\( p^{16} T^{21} + \)\(47\!\cdots\!04\)\( p^{18} T^{22} + \)\(47\!\cdots\!64\)\( p^{21} T^{23} + \)\(65\!\cdots\!33\)\( p^{24} T^{24} + \)\(53\!\cdots\!40\)\( p^{27} T^{25} + 7259804293940170810 p^{30} T^{26} + 4485717947654876 p^{33} T^{27} + 6129207383826 p^{36} T^{28} + 2440833764 p^{39} T^{29} + 3493702 p^{42} T^{30} + 652 p^{45} T^{31} + p^{48} T^{32} \)
79 \( 1 - 1136 T + 3290504 T^{2} - 3102171312 T^{3} + 5343118359048 T^{4} - 4212460707087024 T^{5} + 5696430130488192600 T^{6} - \)\(38\!\cdots\!00\)\( T^{7} + \)\(46\!\cdots\!56\)\( T^{8} - \)\(28\!\cdots\!44\)\( T^{9} + \)\(31\!\cdots\!12\)\( T^{10} - \)\(17\!\cdots\!32\)\( T^{11} + \)\(19\!\cdots\!28\)\( T^{12} - \)\(10\!\cdots\!48\)\( T^{13} + \)\(11\!\cdots\!16\)\( T^{14} - \)\(58\!\cdots\!24\)\( T^{15} + \)\(59\!\cdots\!98\)\( T^{16} - \)\(58\!\cdots\!24\)\( p^{3} T^{17} + \)\(11\!\cdots\!16\)\( p^{6} T^{18} - \)\(10\!\cdots\!48\)\( p^{9} T^{19} + \)\(19\!\cdots\!28\)\( p^{12} T^{20} - \)\(17\!\cdots\!32\)\( p^{15} T^{21} + \)\(31\!\cdots\!12\)\( p^{18} T^{22} - \)\(28\!\cdots\!44\)\( p^{21} T^{23} + \)\(46\!\cdots\!56\)\( p^{24} T^{24} - \)\(38\!\cdots\!00\)\( p^{27} T^{25} + 5696430130488192600 p^{30} T^{26} - 4212460707087024 p^{33} T^{27} + 5343118359048 p^{36} T^{28} - 3102171312 p^{39} T^{29} + 3290504 p^{42} T^{30} - 1136 p^{45} T^{31} + p^{48} T^{32} \)
83 \( 1 + 3300 T + 10551154 T^{2} + 22479136932 T^{3} + 44617340862804 T^{4} + 73062977764417428 T^{5} + \)\(11\!\cdots\!22\)\( T^{6} + \)\(15\!\cdots\!88\)\( T^{7} + \)\(19\!\cdots\!28\)\( T^{8} + \)\(22\!\cdots\!76\)\( T^{9} + \)\(24\!\cdots\!46\)\( T^{10} + \)\(25\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!36\)\( T^{12} + \)\(22\!\cdots\!04\)\( T^{13} + \)\(19\!\cdots\!10\)\( T^{14} + \)\(15\!\cdots\!80\)\( T^{15} + \)\(12\!\cdots\!66\)\( T^{16} + \)\(15\!\cdots\!80\)\( p^{3} T^{17} + \)\(19\!\cdots\!10\)\( p^{6} T^{18} + \)\(22\!\cdots\!04\)\( p^{9} T^{19} + \)\(24\!\cdots\!36\)\( p^{12} T^{20} + \)\(25\!\cdots\!52\)\( p^{15} T^{21} + \)\(24\!\cdots\!46\)\( p^{18} T^{22} + \)\(22\!\cdots\!76\)\( p^{21} T^{23} + \)\(19\!\cdots\!28\)\( p^{24} T^{24} + \)\(15\!\cdots\!88\)\( p^{27} T^{25} + \)\(11\!\cdots\!22\)\( p^{30} T^{26} + 73062977764417428 p^{33} T^{27} + 44617340862804 p^{36} T^{28} + 22479136932 p^{39} T^{29} + 10551154 p^{42} T^{30} + 3300 p^{45} T^{31} + p^{48} T^{32} \)
89 \( 1 + 2416 T + 8658416 T^{2} + 16549834544 T^{3} + 34057100000480 T^{4} + 52817935586084816 T^{5} + 80644224477541831120 T^{6} + \)\(10\!\cdots\!32\)\( T^{7} + \)\(12\!\cdots\!44\)\( T^{8} + \)\(14\!\cdots\!64\)\( T^{9} + \)\(14\!\cdots\!80\)\( T^{10} + \)\(14\!\cdots\!12\)\( T^{11} + \)\(13\!\cdots\!52\)\( T^{12} + \)\(11\!\cdots\!44\)\( T^{13} + \)\(96\!\cdots\!08\)\( T^{14} + \)\(79\!\cdots\!44\)\( T^{15} + \)\(66\!\cdots\!18\)\( T^{16} + \)\(79\!\cdots\!44\)\( p^{3} T^{17} + \)\(96\!\cdots\!08\)\( p^{6} T^{18} + \)\(11\!\cdots\!44\)\( p^{9} T^{19} + \)\(13\!\cdots\!52\)\( p^{12} T^{20} + \)\(14\!\cdots\!12\)\( p^{15} T^{21} + \)\(14\!\cdots\!80\)\( p^{18} T^{22} + \)\(14\!\cdots\!64\)\( p^{21} T^{23} + \)\(12\!\cdots\!44\)\( p^{24} T^{24} + \)\(10\!\cdots\!32\)\( p^{27} T^{25} + 80644224477541831120 p^{30} T^{26} + 52817935586084816 p^{33} T^{27} + 34057100000480 p^{36} T^{28} + 16549834544 p^{39} T^{29} + 8658416 p^{42} T^{30} + 2416 p^{45} T^{31} + p^{48} T^{32} \)
97 \( 1 + 3616 T + 15506868 T^{2} + 38426687008 T^{3} + 98794095726804 T^{4} + 191054549647136160 T^{5} + \)\(37\!\cdots\!96\)\( T^{6} + \)\(59\!\cdots\!92\)\( T^{7} + \)\(95\!\cdots\!08\)\( T^{8} + \)\(13\!\cdots\!12\)\( T^{9} + \)\(18\!\cdots\!60\)\( T^{10} + \)\(22\!\cdots\!52\)\( T^{11} + \)\(27\!\cdots\!80\)\( T^{12} + \)\(30\!\cdots\!84\)\( T^{13} + \)\(33\!\cdots\!28\)\( T^{14} + \)\(33\!\cdots\!48\)\( T^{15} + \)\(33\!\cdots\!74\)\( T^{16} + \)\(33\!\cdots\!48\)\( p^{3} T^{17} + \)\(33\!\cdots\!28\)\( p^{6} T^{18} + \)\(30\!\cdots\!84\)\( p^{9} T^{19} + \)\(27\!\cdots\!80\)\( p^{12} T^{20} + \)\(22\!\cdots\!52\)\( p^{15} T^{21} + \)\(18\!\cdots\!60\)\( p^{18} T^{22} + \)\(13\!\cdots\!12\)\( p^{21} T^{23} + \)\(95\!\cdots\!08\)\( p^{24} T^{24} + \)\(59\!\cdots\!92\)\( p^{27} T^{25} + \)\(37\!\cdots\!96\)\( p^{30} T^{26} + 191054549647136160 p^{33} T^{27} + 98794095726804 p^{36} T^{28} + 38426687008 p^{39} T^{29} + 15506868 p^{42} T^{30} + 3616 p^{45} T^{31} + p^{48} T^{32} \)
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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

−2.55713083323267409150764835512, −2.46684148078146555052671272845, −2.20611417467040612878874864875, −2.10759791365528759944463759770, −2.04285615376017000944507429992, −2.00480660665654635277159167754, −1.92356527848082573519882193980, −1.91128178352395368146925834676, −1.85409944730917236133064683467, −1.72877096203452586449679787632, −1.71153261040758338829610635148, −1.70977734447768732272070052907, −1.60508235462224190135370976272, −1.29545570985379190111442490000, −1.28957328850290534053642282681, −1.21107056740382640596734994596, −1.16663752618927025308289575133, −1.12039988629188490211482412975, −1.08132109300361081851997256653, −1.06977082121457243399768942679, −1.06751906178402665057276910685, −0.986399814140447695002969299084, −0.853451782916020251623180076413, −0.817282821932240496625609900588, −0.69417021123350392705640481699, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.69417021123350392705640481699, 0.817282821932240496625609900588, 0.853451782916020251623180076413, 0.986399814140447695002969299084, 1.06751906178402665057276910685, 1.06977082121457243399768942679, 1.08132109300361081851997256653, 1.12039988629188490211482412975, 1.16663752618927025308289575133, 1.21107056740382640596734994596, 1.28957328850290534053642282681, 1.29545570985379190111442490000, 1.60508235462224190135370976272, 1.70977734447768732272070052907, 1.71153261040758338829610635148, 1.72877096203452586449679787632, 1.85409944730917236133064683467, 1.91128178352395368146925834676, 1.92356527848082573519882193980, 2.00480660665654635277159167754, 2.04285615376017000944507429992, 2.10759791365528759944463759770, 2.20611417467040612878874864875, 2.46684148078146555052671272845, 2.55713083323267409150764835512

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

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