Properties

Label 32-273e16-1.1-c1e16-0-2
Degree $32$
Conductor $9.519\times 10^{38}$
Sign $1$
Analytic cond. $260037.$
Root an. cond. $1.47645$
Motivic weight $1$
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  + 8·3-s − 4-s + 28·9-s − 8·12-s − 12·13-s + 5·16-s − 2·17-s − 6·23-s + 30·25-s + 48·27-s − 12·29-s − 6·32-s − 28·36-s − 6·37-s − 96·39-s − 30·41-s + 14·43-s + 40·48-s + 4·49-s − 16·51-s + 12·52-s + 28·53-s − 24·59-s + 2·61-s − 6·64-s + 30·67-s + 2·68-s + ⋯
L(s)  = 1  + 4.61·3-s − 1/2·4-s + 28/3·9-s − 2.30·12-s − 3.32·13-s + 5/4·16-s − 0.485·17-s − 1.25·23-s + 6·25-s + 9.23·27-s − 2.22·29-s − 1.06·32-s − 4.66·36-s − 0.986·37-s − 15.3·39-s − 4.68·41-s + 2.13·43-s + 5.77·48-s + 4/7·49-s − 2.24·51-s + 1.66·52-s + 3.84·53-s − 3.12·59-s + 0.256·61-s − 3/4·64-s + 3.66·67-s + 0.242·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 7^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(260037.\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{273} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.263473398\)
\(L(\frac12)\) \(\approx\) \(5.263473398\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{8} \)
7 \( ( 1 - T^{2} + T^{4} )^{4} \)
13 \( 1 + 12 T + 87 T^{2} + 36 p T^{3} + 1971 T^{4} + 6738 T^{5} + 18453 T^{6} + 41646 T^{7} + 110993 T^{8} + 41646 p T^{9} + 18453 p^{2} T^{10} + 6738 p^{3} T^{11} + 1971 p^{4} T^{12} + 36 p^{6} T^{13} + 87 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
good2 \( 1 + T^{2} - p^{2} T^{4} + 3 p T^{5} - 3 T^{6} + 3 p^{2} T^{7} - 9 p T^{9} + 3 p T^{10} - 3 p^{4} T^{11} + 79 T^{12} + 9 p^{3} T^{13} + 17 p^{2} T^{14} + 21 p^{2} T^{15} - 247 T^{16} + 21 p^{3} T^{17} + 17 p^{4} T^{18} + 9 p^{6} T^{19} + 79 p^{4} T^{20} - 3 p^{9} T^{21} + 3 p^{7} T^{22} - 9 p^{8} T^{23} + 3 p^{11} T^{25} - 3 p^{10} T^{26} + 3 p^{12} T^{27} - p^{14} T^{28} + p^{14} T^{30} + p^{16} T^{32} \)
5 \( 1 - 6 p T^{2} + 443 T^{4} - 4276 T^{6} + 6301 p T^{8} - 203174 T^{10} + 1263211 T^{12} - 7525488 T^{14} + 40314529 T^{16} - 7525488 p^{2} T^{18} + 1263211 p^{4} T^{20} - 203174 p^{6} T^{22} + 6301 p^{9} T^{24} - 4276 p^{10} T^{26} + 443 p^{12} T^{28} - 6 p^{15} T^{30} + p^{16} T^{32} \)
11 \( 1 + 20 T^{2} + 171 T^{4} - 318 T^{5} - 2362 T^{6} + 3534 T^{7} - 57733 T^{8} + 143502 T^{9} - 362522 T^{10} + 2412444 T^{11} + 567417 T^{12} - 4761456 T^{13} + 51307768 T^{14} - 433232568 T^{15} + 375326845 T^{16} - 433232568 p T^{17} + 51307768 p^{2} T^{18} - 4761456 p^{3} T^{19} + 567417 p^{4} T^{20} + 2412444 p^{5} T^{21} - 362522 p^{6} T^{22} + 143502 p^{7} T^{23} - 57733 p^{8} T^{24} + 3534 p^{9} T^{25} - 2362 p^{10} T^{26} - 318 p^{11} T^{27} + 171 p^{12} T^{28} + 20 p^{14} T^{30} + p^{16} T^{32} \)
17 \( 1 + 2 T - 41 T^{2} - 294 T^{3} + 410 T^{4} + 10276 T^{5} + 29791 T^{6} - 156214 T^{7} - 1204593 T^{8} - 974078 T^{9} + 21633218 T^{10} + 86511284 T^{11} - 86539833 T^{12} - 1854440930 T^{13} - 4715816279 T^{14} + 13819057390 T^{15} + 132905859300 T^{16} + 13819057390 p T^{17} - 4715816279 p^{2} T^{18} - 1854440930 p^{3} T^{19} - 86539833 p^{4} T^{20} + 86511284 p^{5} T^{21} + 21633218 p^{6} T^{22} - 974078 p^{7} T^{23} - 1204593 p^{8} T^{24} - 156214 p^{9} T^{25} + 29791 p^{10} T^{26} + 10276 p^{11} T^{27} + 410 p^{12} T^{28} - 294 p^{13} T^{29} - 41 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \)
19 \( 1 + 73 T^{2} + 2948 T^{4} + 3252 T^{5} + 83129 T^{6} + 168408 T^{7} + 96403 p T^{8} + 4550772 T^{9} + 39248064 T^{10} + 73946484 T^{11} + 786856534 T^{12} + 634369176 T^{13} + 14674043478 T^{14} + 3299885748 T^{15} + 274620956664 T^{16} + 3299885748 p T^{17} + 14674043478 p^{2} T^{18} + 634369176 p^{3} T^{19} + 786856534 p^{4} T^{20} + 73946484 p^{5} T^{21} + 39248064 p^{6} T^{22} + 4550772 p^{7} T^{23} + 96403 p^{9} T^{24} + 168408 p^{9} T^{25} + 83129 p^{10} T^{26} + 3252 p^{11} T^{27} + 2948 p^{12} T^{28} + 73 p^{14} T^{30} + p^{16} T^{32} \)
23 \( 1 + 6 T - 91 T^{2} - 110 T^{3} + 7457 T^{4} - 9202 T^{5} - 318656 T^{6} + 1428804 T^{7} + 9322934 T^{8} - 137844 p^{2} T^{9} - 85790918 T^{10} + 2524398078 T^{11} - 4220508830 T^{12} - 52607690920 T^{13} + 270181836017 T^{14} + 22849971282 p T^{15} - 334573040449 p T^{16} + 22849971282 p^{2} T^{17} + 270181836017 p^{2} T^{18} - 52607690920 p^{3} T^{19} - 4220508830 p^{4} T^{20} + 2524398078 p^{5} T^{21} - 85790918 p^{6} T^{22} - 137844 p^{9} T^{23} + 9322934 p^{8} T^{24} + 1428804 p^{9} T^{25} - 318656 p^{10} T^{26} - 9202 p^{11} T^{27} + 7457 p^{12} T^{28} - 110 p^{13} T^{29} - 91 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
29 \( 1 + 12 T + 18 T^{2} - 376 T^{3} - 2161 T^{4} + 1840 T^{5} + 103824 T^{6} + 520926 T^{7} - 1572752 T^{8} - 17528284 T^{9} + 18821586 T^{10} + 544344096 T^{11} + 2161393422 T^{12} - 1395042284 T^{13} - 60229103551 T^{14} + 28609096538 T^{15} + 2256146434858 T^{16} + 28609096538 p T^{17} - 60229103551 p^{2} T^{18} - 1395042284 p^{3} T^{19} + 2161393422 p^{4} T^{20} + 544344096 p^{5} T^{21} + 18821586 p^{6} T^{22} - 17528284 p^{7} T^{23} - 1572752 p^{8} T^{24} + 520926 p^{9} T^{25} + 103824 p^{10} T^{26} + 1840 p^{11} T^{27} - 2161 p^{12} T^{28} - 376 p^{13} T^{29} + 18 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \)
31 \( 1 - 258 T^{2} + 33109 T^{4} - 2840472 T^{6} + 184703268 T^{8} - 9745394772 T^{10} + 433512768044 T^{12} - 16589494630446 T^{14} + 551196695485436 T^{16} - 16589494630446 p^{2} T^{18} + 433512768044 p^{4} T^{20} - 9745394772 p^{6} T^{22} + 184703268 p^{8} T^{24} - 2840472 p^{10} T^{26} + 33109 p^{12} T^{28} - 258 p^{14} T^{30} + p^{16} T^{32} \)
37 \( 1 + 6 T + 177 T^{2} + 990 T^{3} + 14228 T^{4} + 57474 T^{5} + 700051 T^{6} + 1451718 T^{7} + 27383363 T^{8} + 40001820 T^{9} + 1268583512 T^{10} + 2809915104 T^{11} + 61371023941 T^{12} + 107556285678 T^{13} + 2179482108429 T^{14} + 1787110561974 T^{15} + 69771935069668 T^{16} + 1787110561974 p T^{17} + 2179482108429 p^{2} T^{18} + 107556285678 p^{3} T^{19} + 61371023941 p^{4} T^{20} + 2809915104 p^{5} T^{21} + 1268583512 p^{6} T^{22} + 40001820 p^{7} T^{23} + 27383363 p^{8} T^{24} + 1451718 p^{9} T^{25} + 700051 p^{10} T^{26} + 57474 p^{11} T^{27} + 14228 p^{12} T^{28} + 990 p^{13} T^{29} + 177 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
41 \( 1 + 30 T + 575 T^{2} + 8250 T^{3} + 95739 T^{4} + 943848 T^{5} + 8050892 T^{6} + 60437796 T^{7} + 240335 p^{2} T^{8} + 2433000642 T^{9} + 13621179079 T^{10} + 74819963034 T^{11} + 440654529114 T^{12} + 2931357770826 T^{13} + 21090258657235 T^{14} + 150632286782634 T^{15} + 1006165188471556 T^{16} + 150632286782634 p T^{17} + 21090258657235 p^{2} T^{18} + 2931357770826 p^{3} T^{19} + 440654529114 p^{4} T^{20} + 74819963034 p^{5} T^{21} + 13621179079 p^{6} T^{22} + 2433000642 p^{7} T^{23} + 240335 p^{10} T^{24} + 60437796 p^{9} T^{25} + 8050892 p^{10} T^{26} + 943848 p^{11} T^{27} + 95739 p^{12} T^{28} + 8250 p^{13} T^{29} + 575 p^{14} T^{30} + 30 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 - 14 T - 25 T^{2} + 98 T^{3} + 12097 T^{4} - 22968 T^{5} - 305704 T^{6} - 3814866 T^{7} + 12100128 T^{8} + 118230986 T^{9} + 1147656572 T^{10} - 1958074954 T^{11} - 50432478520 T^{12} - 244451897020 T^{13} + 306361524951 T^{14} + 8746158934314 T^{15} + 26545402320665 T^{16} + 8746158934314 p T^{17} + 306361524951 p^{2} T^{18} - 244451897020 p^{3} T^{19} - 50432478520 p^{4} T^{20} - 1958074954 p^{5} T^{21} + 1147656572 p^{6} T^{22} + 118230986 p^{7} T^{23} + 12100128 p^{8} T^{24} - 3814866 p^{9} T^{25} - 305704 p^{10} T^{26} - 22968 p^{11} T^{27} + 12097 p^{12} T^{28} + 98 p^{13} T^{29} - 25 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \)
47 \( 1 - 186 T^{2} + 13601 T^{4} - 661588 T^{6} + 40610180 T^{8} - 2326785416 T^{10} + 99861652240 T^{12} - 5727716857806 T^{14} + 335125347067804 T^{16} - 5727716857806 p^{2} T^{18} + 99861652240 p^{4} T^{20} - 2326785416 p^{6} T^{22} + 40610180 p^{8} T^{24} - 661588 p^{10} T^{26} + 13601 p^{12} T^{28} - 186 p^{14} T^{30} + p^{16} T^{32} \)
53 \( ( 1 - 14 T + 300 T^{2} - 2760 T^{3} + 32762 T^{4} - 198222 T^{5} + 1772024 T^{6} - 7335806 T^{7} + 77314059 T^{8} - 7335806 p T^{9} + 1772024 p^{2} T^{10} - 198222 p^{3} T^{11} + 32762 p^{4} T^{12} - 2760 p^{5} T^{13} + 300 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( 1 + 24 T + 539 T^{2} + 8328 T^{3} + 115251 T^{4} + 1326900 T^{5} + 14050742 T^{6} + 129142968 T^{7} + 1111014044 T^{8} + 8281314384 T^{9} + 56882945752 T^{10} + 309726060258 T^{11} + 1283893911360 T^{12} - 1007575966098 T^{13} - 71532433148903 T^{14} - 980486075231754 T^{15} - 8181376692863111 T^{16} - 980486075231754 p T^{17} - 71532433148903 p^{2} T^{18} - 1007575966098 p^{3} T^{19} + 1283893911360 p^{4} T^{20} + 309726060258 p^{5} T^{21} + 56882945752 p^{6} T^{22} + 8281314384 p^{7} T^{23} + 1111014044 p^{8} T^{24} + 129142968 p^{9} T^{25} + 14050742 p^{10} T^{26} + 1326900 p^{11} T^{27} + 115251 p^{12} T^{28} + 8328 p^{13} T^{29} + 539 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \)
61 \( 1 - 2 T - 348 T^{2} + 528 T^{3} + 62122 T^{4} - 67978 T^{5} - 8083588 T^{6} + 6720190 T^{7} + 871782213 T^{8} - 567006886 T^{9} - 80756580288 T^{10} + 36167132182 T^{11} + 6542766246442 T^{12} - 1582215717876 T^{13} - 471197353489472 T^{14} + 34085218272410 T^{15} + 30371717222165356 T^{16} + 34085218272410 p T^{17} - 471197353489472 p^{2} T^{18} - 1582215717876 p^{3} T^{19} + 6542766246442 p^{4} T^{20} + 36167132182 p^{5} T^{21} - 80756580288 p^{6} T^{22} - 567006886 p^{7} T^{23} + 871782213 p^{8} T^{24} + 6720190 p^{9} T^{25} - 8083588 p^{10} T^{26} - 67978 p^{11} T^{27} + 62122 p^{12} T^{28} + 528 p^{13} T^{29} - 348 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 - 30 T + 735 T^{2} - 13050 T^{3} + 203725 T^{4} - 2666316 T^{5} + 31653972 T^{6} - 334495446 T^{7} + 3283743828 T^{8} - 29916484650 T^{9} + 262104541728 T^{10} - 2242119948858 T^{11} + 19092620081732 T^{12} - 164189360358072 T^{13} + 1404467812435059 T^{14} - 11974380158039754 T^{15} + 98830332492887945 T^{16} - 11974380158039754 p T^{17} + 1404467812435059 p^{2} T^{18} - 164189360358072 p^{3} T^{19} + 19092620081732 p^{4} T^{20} - 2242119948858 p^{5} T^{21} + 262104541728 p^{6} T^{22} - 29916484650 p^{7} T^{23} + 3283743828 p^{8} T^{24} - 334495446 p^{9} T^{25} + 31653972 p^{10} T^{26} - 2666316 p^{11} T^{27} + 203725 p^{12} T^{28} - 13050 p^{13} T^{29} + 735 p^{14} T^{30} - 30 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 + 6 T + 344 T^{2} + 1992 T^{3} + 60465 T^{4} + 284676 T^{5} + 6504830 T^{6} + 22932720 T^{7} + 460917815 T^{8} + 1130102280 T^{9} + 23713389340 T^{10} + 84334955376 T^{11} + 1329013275657 T^{12} + 13430256823806 T^{13} + 119003779607140 T^{14} + 1583719176585852 T^{15} + 9915173847443161 T^{16} + 1583719176585852 p T^{17} + 119003779607140 p^{2} T^{18} + 13430256823806 p^{3} T^{19} + 1329013275657 p^{4} T^{20} + 84334955376 p^{5} T^{21} + 23713389340 p^{6} T^{22} + 1130102280 p^{7} T^{23} + 460917815 p^{8} T^{24} + 22932720 p^{9} T^{25} + 6504830 p^{10} T^{26} + 284676 p^{11} T^{27} + 60465 p^{12} T^{28} + 1992 p^{13} T^{29} + 344 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 - 618 T^{2} + 196687 T^{4} - 42567516 T^{6} + 6980445477 T^{8} - 917847146298 T^{10} + 100043030404079 T^{12} - 9221904620581356 T^{14} + 726792737728641401 T^{16} - 9221904620581356 p^{2} T^{18} + 100043030404079 p^{4} T^{20} - 917847146298 p^{6} T^{22} + 6980445477 p^{8} T^{24} - 42567516 p^{10} T^{26} + 196687 p^{12} T^{28} - 618 p^{14} T^{30} + p^{16} T^{32} \)
79 \( ( 1 - 46 T + 1373 T^{2} - 29676 T^{3} + 518304 T^{4} - 7532304 T^{5} + 94017544 T^{6} - 1017546494 T^{7} + 9661278440 T^{8} - 1017546494 p T^{9} + 94017544 p^{2} T^{10} - 7532304 p^{3} T^{11} + 518304 p^{4} T^{12} - 29676 p^{5} T^{13} + 1373 p^{6} T^{14} - 46 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( 1 - 756 T^{2} + 269174 T^{4} - 60833818 T^{6} + 9951538541 T^{8} - 1276173252560 T^{10} + 136102909547275 T^{12} - 12723814226688534 T^{14} + 1091530395618435706 T^{16} - 12723814226688534 p^{2} T^{18} + 136102909547275 p^{4} T^{20} - 1276173252560 p^{6} T^{22} + 9951538541 p^{8} T^{24} - 60833818 p^{10} T^{26} + 269174 p^{12} T^{28} - 756 p^{14} T^{30} + p^{16} T^{32} \)
89 \( 1 - 18 T + 501 T^{2} - 7074 T^{3} + 120388 T^{4} - 1606998 T^{5} + 22974801 T^{6} - 291334410 T^{7} + 3686800389 T^{8} - 42582618516 T^{9} + 491883376368 T^{10} - 5340094232172 T^{11} + 58241112783722 T^{12} - 598116184864884 T^{13} + 6127033409533722 T^{14} - 58887883494753720 T^{15} + 572157617159140568 T^{16} - 58887883494753720 p T^{17} + 6127033409533722 p^{2} T^{18} - 598116184864884 p^{3} T^{19} + 58241112783722 p^{4} T^{20} - 5340094232172 p^{5} T^{21} + 491883376368 p^{6} T^{22} - 42582618516 p^{7} T^{23} + 3686800389 p^{8} T^{24} - 291334410 p^{9} T^{25} + 22974801 p^{10} T^{26} - 1606998 p^{11} T^{27} + 120388 p^{12} T^{28} - 7074 p^{13} T^{29} + 501 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \)
97 \( 1 + 6 T + 547 T^{2} + 3210 T^{3} + 165383 T^{4} + 808812 T^{5} + 32896490 T^{6} + 106625022 T^{7} + 4598294392 T^{8} + 1440052458 T^{9} + 442215729636 T^{10} - 2528804866020 T^{11} + 24902565584716 T^{12} - 614588315539890 T^{13} + 62333990757309 T^{14} - 86521863369201144 T^{15} - 100749508181265495 T^{16} - 86521863369201144 p T^{17} + 62333990757309 p^{2} T^{18} - 614588315539890 p^{3} T^{19} + 24902565584716 p^{4} T^{20} - 2528804866020 p^{5} T^{21} + 442215729636 p^{6} T^{22} + 1440052458 p^{7} T^{23} + 4598294392 p^{8} T^{24} + 106625022 p^{9} T^{25} + 32896490 p^{10} T^{26} + 808812 p^{11} T^{27} + 165383 p^{12} T^{28} + 3210 p^{13} T^{29} + 547 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
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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

−3.43416963417695317603139617240, −3.36463871985084796816680617486, −3.33944192466494705197134553226, −3.11055545909493115274141146038, −3.06672338010932645595048719475, −3.01517120750339124176614306717, −2.72697729618634814042597632130, −2.67669109315649238727974834335, −2.55837453557275015142563718932, −2.45615488697700526304118161982, −2.44775914028182781655924382732, −2.35216438680524340918771762880, −2.31456796659939066909557379335, −2.30396450128063759905625856705, −2.12430017928522832265562381455, −2.08243932938403338869906734902, −1.98751999059164158085736050719, −1.81149533727705152539289343249, −1.56212433082419475863969869174, −1.42153645387276627944332598064, −1.30027106143439142890700495617, −1.04955694651707827952481344434, −0.888676111604621731149161307124, −0.73181851413728863054185296625, −0.18461574104759830359332073419, 0.18461574104759830359332073419, 0.73181851413728863054185296625, 0.888676111604621731149161307124, 1.04955694651707827952481344434, 1.30027106143439142890700495617, 1.42153645387276627944332598064, 1.56212433082419475863969869174, 1.81149533727705152539289343249, 1.98751999059164158085736050719, 2.08243932938403338869906734902, 2.12430017928522832265562381455, 2.30396450128063759905625856705, 2.31456796659939066909557379335, 2.35216438680524340918771762880, 2.44775914028182781655924382732, 2.45615488697700526304118161982, 2.55837453557275015142563718932, 2.67669109315649238727974834335, 2.72697729618634814042597632130, 3.01517120750339124176614306717, 3.06672338010932645595048719475, 3.11055545909493115274141146038, 3.33944192466494705197134553226, 3.36463871985084796816680617486, 3.43416963417695317603139617240

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

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