Properties

Label 32-1110e16-1.1-c1e16-0-0
Degree $32$
Conductor $5.311\times 10^{48}$
Sign $1$
Analytic cond. $1.45077\times 10^{15}$
Root an. cond. $2.97714$
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 + 4·4-s − 2·7-s + 28·9-s − 8·11-s − 32·12-s − 6·13-s + 6·16-s + 6·17-s − 12·19-s + 16·21-s + 4·25-s − 48·27-s − 8·28-s + 64·33-s + 112·36-s + 12·37-s + 48·39-s + 4·41-s − 32·44-s + 68·47-s − 48·48-s + 28·49-s − 48·51-s − 24·52-s − 12·53-s + 96·57-s + ⋯
L(s)  = 1  − 4.61·3-s + 2·4-s − 0.755·7-s + 28/3·9-s − 2.41·11-s − 9.23·12-s − 1.66·13-s + 3/2·16-s + 1.45·17-s − 2.75·19-s + 3.49·21-s + 4/5·25-s − 9.23·27-s − 1.51·28-s + 11.1·33-s + 56/3·36-s + 1.97·37-s + 7.68·39-s + 0.624·41-s − 4.82·44-s + 9.91·47-s − 6.92·48-s + 4·49-s − 6.72·51-s − 3.32·52-s − 1.64·53-s + 12.7·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 37^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 37^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 37^{16}\)
Sign: $1$
Analytic conductor: \(1.45077\times 10^{15}\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1110} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 37^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.005189276646\)
\(L(\frac12)\) \(\approx\) \(0.005189276646\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{4} \)
3 \( ( 1 + T + T^{2} )^{8} \)
5 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( 1 - 12 T + 51 T^{2} + 200 T^{3} - 4031 T^{4} + 16292 T^{5} + 1750 T^{6} - 270836 T^{7} + 2319866 T^{8} - 270836 p T^{9} + 1750 p^{2} T^{10} + 16292 p^{3} T^{11} - 4031 p^{4} T^{12} + 200 p^{5} T^{13} + 51 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
good7 \( 1 + 2 T - 24 T^{2} - 76 T^{3} + 33 p T^{4} + 24 p^{2} T^{5} - 708 T^{6} - 10170 T^{7} - 6731 T^{8} + 57396 T^{9} + 107404 T^{10} - 240748 T^{11} - 977206 T^{12} + 117676 p T^{13} + 1126788 p T^{14} - 35676 p^{2} T^{15} - 1183718 p^{2} T^{16} - 35676 p^{3} T^{17} + 1126788 p^{3} T^{18} + 117676 p^{4} T^{19} - 977206 p^{4} T^{20} - 240748 p^{5} T^{21} + 107404 p^{6} T^{22} + 57396 p^{7} T^{23} - 6731 p^{8} T^{24} - 10170 p^{9} T^{25} - 708 p^{10} T^{26} + 24 p^{13} T^{27} + 33 p^{13} T^{28} - 76 p^{13} T^{29} - 24 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \)
11 \( ( 1 + 4 T + 26 T^{2} + 76 T^{3} + 521 T^{4} + 1376 T^{5} + 7210 T^{6} + 18600 T^{7} + 96804 T^{8} + 18600 p T^{9} + 7210 p^{2} T^{10} + 1376 p^{3} T^{11} + 521 p^{4} T^{12} + 76 p^{5} T^{13} + 26 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
13 \( 1 + 6 T + 48 T^{2} + 216 T^{3} + 74 p T^{4} + 3810 T^{5} + 11308 T^{6} + 34410 T^{7} + 49245 T^{8} - 33102 T^{9} - 750688 T^{10} - 3291066 T^{11} - 5674750 T^{12} + 14134776 T^{13} + 193406660 T^{14} + 1022859774 T^{15} + 4299743068 T^{16} + 1022859774 p T^{17} + 193406660 p^{2} T^{18} + 14134776 p^{3} T^{19} - 5674750 p^{4} T^{20} - 3291066 p^{5} T^{21} - 750688 p^{6} T^{22} - 33102 p^{7} T^{23} + 49245 p^{8} T^{24} + 34410 p^{9} T^{25} + 11308 p^{10} T^{26} + 3810 p^{11} T^{27} + 74 p^{13} T^{28} + 216 p^{13} T^{29} + 48 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
17 \( 1 - 6 T + 122 T^{2} - 660 T^{3} + 7631 T^{4} - 39732 T^{5} + 340506 T^{6} - 100326 p T^{7} + 12019985 T^{8} - 56964672 T^{9} + 350019296 T^{10} - 1559629728 T^{11} + 8622411038 T^{12} - 36055172916 T^{13} + 182065577580 T^{14} - 712587653976 T^{15} + 3322595918802 T^{16} - 712587653976 p T^{17} + 182065577580 p^{2} T^{18} - 36055172916 p^{3} T^{19} + 8622411038 p^{4} T^{20} - 1559629728 p^{5} T^{21} + 350019296 p^{6} T^{22} - 56964672 p^{7} T^{23} + 12019985 p^{8} T^{24} - 100326 p^{10} T^{25} + 340506 p^{10} T^{26} - 39732 p^{11} T^{27} + 7631 p^{12} T^{28} - 660 p^{13} T^{29} + 122 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
19 \( 1 + 12 T + 131 T^{2} + 996 T^{3} + 6763 T^{4} + 37896 T^{5} + 194160 T^{6} + 866736 T^{7} + 3601891 T^{8} + 13500156 T^{9} + 48990343 T^{10} + 170273100 T^{11} + 645427170 T^{12} + 2562258372 T^{13} + 11723832803 T^{14} + 51957105444 T^{15} + 238616673728 T^{16} + 51957105444 p T^{17} + 11723832803 p^{2} T^{18} + 2562258372 p^{3} T^{19} + 645427170 p^{4} T^{20} + 170273100 p^{5} T^{21} + 48990343 p^{6} T^{22} + 13500156 p^{7} T^{23} + 3601891 p^{8} T^{24} + 866736 p^{9} T^{25} + 194160 p^{10} T^{26} + 37896 p^{11} T^{27} + 6763 p^{12} T^{28} + 996 p^{13} T^{29} + 131 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \)
23 \( 1 - 208 T^{2} + 21132 T^{4} - 1395808 T^{6} + 67592772 T^{8} - 2577232384 T^{10} + 81487897204 T^{12} - 2225453116880 T^{14} + 54011563159798 T^{16} - 2225453116880 p^{2} T^{18} + 81487897204 p^{4} T^{20} - 2577232384 p^{6} T^{22} + 67592772 p^{8} T^{24} - 1395808 p^{10} T^{26} + 21132 p^{12} T^{28} - 208 p^{14} T^{30} + p^{16} T^{32} \)
29 \( 1 - 242 T^{2} + 30291 T^{4} - 2574518 T^{6} + 165669577 T^{8} - 8550371196 T^{10} + 366462797006 T^{12} - 13326890009056 T^{14} + 416225405397090 T^{16} - 13326890009056 p^{2} T^{18} + 366462797006 p^{4} T^{20} - 8550371196 p^{6} T^{22} + 165669577 p^{8} T^{24} - 2574518 p^{10} T^{26} + 30291 p^{12} T^{28} - 242 p^{14} T^{30} + p^{16} T^{32} \)
31 \( 1 - 298 T^{2} + 44077 T^{4} - 4321386 T^{6} + 315553222 T^{8} - 18239633606 T^{10} + 864610341315 T^{12} - 34324232132518 T^{14} + 1154125274406290 T^{16} - 34324232132518 p^{2} T^{18} + 864610341315 p^{4} T^{20} - 18239633606 p^{6} T^{22} + 315553222 p^{8} T^{24} - 4321386 p^{10} T^{26} + 44077 p^{12} T^{28} - 298 p^{14} T^{30} + p^{16} T^{32} \)
41 \( 1 - 4 T - 8 T^{2} - 992 T^{3} + 564 T^{4} + 11016 T^{5} + 644160 T^{6} + 1005052 T^{7} - 929126 T^{8} - 300746640 T^{9} - 956951824 T^{10} - 3123173656 T^{11} + 101361069904 T^{12} + 468452385792 T^{13} + 57841866232 p T^{14} - 24396631999760 T^{15} - 148780782025741 T^{16} - 24396631999760 p T^{17} + 57841866232 p^{3} T^{18} + 468452385792 p^{3} T^{19} + 101361069904 p^{4} T^{20} - 3123173656 p^{5} T^{21} - 956951824 p^{6} T^{22} - 300746640 p^{7} T^{23} - 929126 p^{8} T^{24} + 1005052 p^{9} T^{25} + 644160 p^{10} T^{26} + 11016 p^{11} T^{27} + 564 p^{12} T^{28} - 992 p^{13} T^{29} - 8 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 - 546 T^{2} + 142265 T^{4} - 23550410 T^{6} + 2782967058 T^{8} - 250100351398 T^{10} + 17771727129527 T^{12} - 1023393036711022 T^{14} + 48431640694317850 T^{16} - 1023393036711022 p^{2} T^{18} + 17771727129527 p^{4} T^{20} - 250100351398 p^{6} T^{22} + 2782967058 p^{8} T^{24} - 23550410 p^{10} T^{26} + 142265 p^{12} T^{28} - 546 p^{14} T^{30} + p^{16} T^{32} \)
47 \( ( 1 - 34 T + 720 T^{2} - 10910 T^{3} + 131977 T^{4} - 1332872 T^{5} + 11685026 T^{6} - 91644052 T^{7} + 654884968 T^{8} - 91644052 p T^{9} + 11685026 p^{2} T^{10} - 1332872 p^{3} T^{11} + 131977 p^{4} T^{12} - 10910 p^{5} T^{13} + 720 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
53 \( 1 + 12 T + 16 T^{2} - 1264 T^{3} - 14524 T^{4} - 42560 T^{5} + 916208 T^{6} + 11222796 T^{7} + 36194330 T^{8} - 449089904 T^{9} - 5802269416 T^{10} - 21878066352 T^{11} + 181815924688 T^{12} + 2530801135384 T^{13} + 9478229667584 T^{14} - 70372915927088 T^{15} - 975491825794349 T^{16} - 70372915927088 p T^{17} + 9478229667584 p^{2} T^{18} + 2530801135384 p^{3} T^{19} + 181815924688 p^{4} T^{20} - 21878066352 p^{5} T^{21} - 5802269416 p^{6} T^{22} - 449089904 p^{7} T^{23} + 36194330 p^{8} T^{24} + 11222796 p^{9} T^{25} + 916208 p^{10} T^{26} - 42560 p^{11} T^{27} - 14524 p^{12} T^{28} - 1264 p^{13} T^{29} + 16 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \)
59 \( 1 - 6 T + 272 T^{2} - 1560 T^{3} + 36555 T^{4} - 169320 T^{5} + 2898356 T^{6} - 9277818 T^{7} + 141334569 T^{8} - 200113104 T^{9} + 4786520684 T^{10} - 8398393584 T^{11} + 280320577870 T^{12} - 2541258906588 T^{13} + 30706084050484 T^{14} - 296682539837136 T^{15} + 2328805656167986 T^{16} - 296682539837136 p T^{17} + 30706084050484 p^{2} T^{18} - 2541258906588 p^{3} T^{19} + 280320577870 p^{4} T^{20} - 8398393584 p^{5} T^{21} + 4786520684 p^{6} T^{22} - 200113104 p^{7} T^{23} + 141334569 p^{8} T^{24} - 9277818 p^{9} T^{25} + 2898356 p^{10} T^{26} - 169320 p^{11} T^{27} + 36555 p^{12} T^{28} - 1560 p^{13} T^{29} + 272 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
61 \( 1 - 12 T + 296 T^{2} - 2976 T^{3} + 40372 T^{4} - 360360 T^{5} + 3460128 T^{6} - 29983476 T^{7} + 237031642 T^{8} - 2160918648 T^{9} + 16380639232 T^{10} - 154902958872 T^{11} + 1256981568336 T^{12} - 11236975622400 T^{13} + 96080373213704 T^{14} - 771776450050056 T^{15} + 6432198562305299 T^{16} - 771776450050056 p T^{17} + 96080373213704 p^{2} T^{18} - 11236975622400 p^{3} T^{19} + 1256981568336 p^{4} T^{20} - 154902958872 p^{5} T^{21} + 16380639232 p^{6} T^{22} - 2160918648 p^{7} T^{23} + 237031642 p^{8} T^{24} - 29983476 p^{9} T^{25} + 3460128 p^{10} T^{26} - 360360 p^{11} T^{27} + 40372 p^{12} T^{28} - 2976 p^{13} T^{29} + 296 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 + 36 T + 405 T^{2} + 188 T^{3} - 15004 T^{4} + 195560 T^{5} + 3389445 T^{6} + 7879116 T^{7} - 55425599 T^{8} - 188416804 T^{9} - 1005170496 T^{10} + 29440090332 T^{11} + 421687495758 T^{12} - 4696995907208 T^{13} - 87406949146994 T^{14} - 196070265281796 T^{15} + 2251860084574648 T^{16} - 196070265281796 p T^{17} - 87406949146994 p^{2} T^{18} - 4696995907208 p^{3} T^{19} + 421687495758 p^{4} T^{20} + 29440090332 p^{5} T^{21} - 1005170496 p^{6} T^{22} - 188416804 p^{7} T^{23} - 55425599 p^{8} T^{24} + 7879116 p^{9} T^{25} + 3389445 p^{10} T^{26} + 195560 p^{11} T^{27} - 15004 p^{12} T^{28} + 188 p^{13} T^{29} + 405 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 - 6 T - 335 T^{2} - 74 T^{3} + 74897 T^{4} + 290912 T^{5} - 9836134 T^{6} - 85509396 T^{7} + 838867295 T^{8} + 12822492038 T^{9} - 22668579505 T^{10} - 1317862649226 T^{11} - 4141287243602 T^{12} + 85350517436498 T^{13} + 769154237455571 T^{14} - 2568355357478806 T^{15} - 68139766681311332 T^{16} - 2568355357478806 p T^{17} + 769154237455571 p^{2} T^{18} + 85350517436498 p^{3} T^{19} - 4141287243602 p^{4} T^{20} - 1317862649226 p^{5} T^{21} - 22668579505 p^{6} T^{22} + 12822492038 p^{7} T^{23} + 838867295 p^{8} T^{24} - 85509396 p^{9} T^{25} - 9836134 p^{10} T^{26} + 290912 p^{11} T^{27} + 74897 p^{12} T^{28} - 74 p^{13} T^{29} - 335 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
73 \( ( 1 + 8 T + 331 T^{2} + 1758 T^{3} + 46620 T^{4} + 127398 T^{5} + 3939885 T^{6} + 2071364 T^{7} + 278165974 T^{8} + 2071364 p T^{9} + 3939885 p^{2} T^{10} + 127398 p^{3} T^{11} + 46620 p^{4} T^{12} + 1758 p^{5} T^{13} + 331 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
79 \( 1 + 24 T + 725 T^{2} + 12792 T^{3} + 239605 T^{4} + 3561708 T^{5} + 53038458 T^{6} + 698171052 T^{7} + 8982094015 T^{8} + 106834711044 T^{9} + 1233597079831 T^{10} + 13421835294996 T^{11} + 141955955990946 T^{12} + 1424990982382692 T^{13} + 13960550936189183 T^{14} + 130048865842686084 T^{15} + 1185428783280273080 T^{16} + 130048865842686084 p T^{17} + 13960550936189183 p^{2} T^{18} + 1424990982382692 p^{3} T^{19} + 141955955990946 p^{4} T^{20} + 13421835294996 p^{5} T^{21} + 1233597079831 p^{6} T^{22} + 106834711044 p^{7} T^{23} + 8982094015 p^{8} T^{24} + 698171052 p^{9} T^{25} + 53038458 p^{10} T^{26} + 3561708 p^{11} T^{27} + 239605 p^{12} T^{28} + 12792 p^{13} T^{29} + 725 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \)
83 \( 1 - 12 T - 227 T^{2} + 3032 T^{3} + 22469 T^{4} - 287768 T^{5} - 1906630 T^{6} + 6921276 T^{7} + 236339219 T^{8} + 883539256 T^{9} - 17707390813 T^{10} - 173989526184 T^{11} + 420684050110 T^{12} + 20723278555624 T^{13} - 16774866286345 T^{14} - 942517472496752 T^{15} + 4007233628230216 T^{16} - 942517472496752 p T^{17} - 16774866286345 p^{2} T^{18} + 20723278555624 p^{3} T^{19} + 420684050110 p^{4} T^{20} - 173989526184 p^{5} T^{21} - 17707390813 p^{6} T^{22} + 883539256 p^{7} T^{23} + 236339219 p^{8} T^{24} + 6921276 p^{9} T^{25} - 1906630 p^{10} T^{26} - 287768 p^{11} T^{27} + 22469 p^{12} T^{28} + 3032 p^{13} T^{29} - 227 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \)
89 \( 1 - 24 T + 596 T^{2} - 9696 T^{3} + 151836 T^{4} - 1753464 T^{5} + 18592568 T^{6} - 141488088 T^{7} + 741372042 T^{8} + 2273640720 T^{9} - 97667751124 T^{10} + 1384893226152 T^{11} - 13428467844464 T^{12} + 83625335358120 T^{13} - 244521931110644 T^{14} - 3049031246590128 T^{15} + 39857338162602979 T^{16} - 3049031246590128 p T^{17} - 244521931110644 p^{2} T^{18} + 83625335358120 p^{3} T^{19} - 13428467844464 p^{4} T^{20} + 1384893226152 p^{5} T^{21} - 97667751124 p^{6} T^{22} + 2273640720 p^{7} T^{23} + 741372042 p^{8} T^{24} - 141488088 p^{9} T^{25} + 18592568 p^{10} T^{26} - 1753464 p^{11} T^{27} + 151836 p^{12} T^{28} - 9696 p^{13} T^{29} + 596 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \)
97 \( 1 - 1282 T^{2} + 783313 T^{4} - 303542706 T^{6} + 83737330546 T^{8} - 17491991471486 T^{10} + 2870945226650703 T^{12} - 378675955035770062 T^{14} + 40639998666074131418 T^{16} - 378675955035770062 p^{2} T^{18} + 2870945226650703 p^{4} T^{20} - 17491991471486 p^{6} T^{22} + 83737330546 p^{8} T^{24} - 303542706 p^{10} T^{26} + 783313 p^{12} T^{28} - 1282 p^{14} T^{30} + p^{16} T^{32} \)
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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

−2.49337603451683394348889396007, −2.44205441632028104908802624983, −2.44104215565122139786347283566, −2.42519511659805499858699691931, −2.38541602757994576616269493022, −2.34802342547754545347732284620, −2.16174851244368822205487184355, −2.04698492328620201482504693639, −1.99226198316390518190973317740, −1.68824892454472151296082299957, −1.57262685467464723290722826836, −1.53124206254629286973398831867, −1.50658029193892997995906139159, −1.45452347008478091322773063670, −1.23386513455337320660632807960, −1.01312276703569242280328435318, −1.00945928396299996330359097864, −0.993130458707182892315127290757, −0.961064926750539752848348171426, −0.935980435007520130971070935246, −0.72863556826476402441704807177, −0.37430314838962755349150649676, −0.21349065529766710323582832363, −0.11118236243817074867446417338, −0.05015036711105207806542430430, 0.05015036711105207806542430430, 0.11118236243817074867446417338, 0.21349065529766710323582832363, 0.37430314838962755349150649676, 0.72863556826476402441704807177, 0.935980435007520130971070935246, 0.961064926750539752848348171426, 0.993130458707182892315127290757, 1.00945928396299996330359097864, 1.01312276703569242280328435318, 1.23386513455337320660632807960, 1.45452347008478091322773063670, 1.50658029193892997995906139159, 1.53124206254629286973398831867, 1.57262685467464723290722826836, 1.68824892454472151296082299957, 1.99226198316390518190973317740, 2.04698492328620201482504693639, 2.16174851244368822205487184355, 2.34802342547754545347732284620, 2.38541602757994576616269493022, 2.42519511659805499858699691931, 2.44104215565122139786347283566, 2.44205441632028104908802624983, 2.49337603451683394348889396007

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

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