Dirichlet series
| L(s) = 1 | + 4·4-s − 12·11-s + 10·13-s + 6·16-s − 6·17-s − 12·23-s + 30·25-s + 16·29-s − 6·37-s + 12·41-s − 6·43-s − 48·44-s + 4·49-s + 40·52-s − 40·53-s + 6·59-s − 2·61-s − 30·67-s − 24·68-s + 12·71-s − 16·79-s + 30·89-s − 48·92-s − 24·97-s + 120·100-s − 48·101-s − 8·107-s + ⋯ |
| L(s) = 1 | + 2·4-s − 3.61·11-s + 2.77·13-s + 3/2·16-s − 1.45·17-s − 2.50·23-s + 6·25-s + 2.97·29-s − 0.986·37-s + 1.87·41-s − 0.914·43-s − 7.23·44-s + 4/7·49-s + 5.54·52-s − 5.49·53-s + 0.781·59-s − 0.256·61-s − 3.66·67-s − 2.91·68-s + 1.42·71-s − 1.80·79-s + 3.17·89-s − 5.00·92-s − 2.43·97-s + 12·100-s − 4.77·101-s − 0.773·107-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.33594\times 10^{17}\) |
| Root analytic conductor: | \(3.61655\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(21.94219939\) |
| \(L(\frac12)\) | \(\approx\) | \(21.94219939\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 - T^{2} + T^{4} )^{4} \) | |
| 13 | \( 1 - 10 T + 32 T^{2} - 4 p T^{3} - 18 T^{4} + 1114 T^{5} - 568 T^{6} - 21158 T^{7} + 93063 T^{8} - 21158 p T^{9} - 568 p^{2} T^{10} + 1114 p^{3} T^{11} - 18 p^{4} T^{12} - 4 p^{6} T^{13} + 32 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 - 6 p T^{2} + 453 T^{4} - 4678 T^{6} + 38174 T^{8} - 53282 p T^{10} + 330287 p T^{12} - 9283762 T^{14} + 48139394 T^{16} - 9283762 p^{2} T^{18} + 330287 p^{5} T^{20} - 53282 p^{7} T^{22} + 38174 p^{8} T^{24} - 4678 p^{10} T^{26} + 453 p^{12} T^{28} - 6 p^{15} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3871 T^{4} + 18912 T^{5} + 83830 T^{6} + 339612 T^{7} + 10465 p^{2} T^{8} + 4283160 T^{9} + 13028660 T^{10} + 34167768 T^{11} + 71980350 T^{12} + 704928 p^{2} T^{13} - 164034664 T^{14} - 12928248 p^{2} T^{15} - 6378725174 T^{16} - 12928248 p^{3} T^{17} - 164034664 p^{2} T^{18} + 704928 p^{5} T^{19} + 71980350 p^{4} T^{20} + 34167768 p^{5} T^{21} + 13028660 p^{6} T^{22} + 4283160 p^{7} T^{23} + 10465 p^{10} T^{24} + 339612 p^{9} T^{25} + 83830 p^{10} T^{26} + 18912 p^{11} T^{27} + 3871 p^{12} T^{28} + 696 p^{13} T^{29} + 106 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 6 T - 28 T^{2} - 296 T^{3} - 22 p T^{4} + 250 p T^{5} + 18004 T^{6} - 11070 T^{7} - 144835 T^{8} + 25074 p T^{9} + 1617800 T^{10} - 17089722 T^{11} - 81807558 T^{12} + 130338856 T^{13} + 1651144512 T^{14} + 202742630 T^{15} - 24526106756 T^{16} + 202742630 p T^{17} + 1651144512 p^{2} T^{18} + 130338856 p^{3} T^{19} - 81807558 p^{4} T^{20} - 17089722 p^{5} T^{21} + 1617800 p^{6} T^{22} + 25074 p^{8} T^{23} - 144835 p^{8} T^{24} - 11070 p^{9} T^{25} + 18004 p^{10} T^{26} + 250 p^{12} T^{27} - 22 p^{13} T^{28} - 296 p^{13} T^{29} - 28 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 90 T^{2} + 4271 T^{4} - 1248 T^{5} + 144062 T^{6} - 150048 T^{7} + 3788121 T^{8} - 8725728 T^{9} + 80033644 T^{10} - 338065248 T^{11} + 74080682 p T^{12} - 9944408736 T^{13} + 22511789392 T^{14} - 233454461280 T^{15} + 388714746634 T^{16} - 233454461280 p T^{17} + 22511789392 p^{2} T^{18} - 9944408736 p^{3} T^{19} + 74080682 p^{5} T^{20} - 338065248 p^{5} T^{21} + 80033644 p^{6} T^{22} - 8725728 p^{7} T^{23} + 3788121 p^{8} T^{24} - 150048 p^{9} T^{25} + 144062 p^{10} T^{26} - 1248 p^{11} T^{27} + 4271 p^{12} T^{28} + 90 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 12 T + 7 T^{2} - 316 T^{3} - 637 T^{4} - 2984 T^{5} - 55264 T^{6} - 174480 T^{7} + 732467 T^{8} + 7902460 T^{9} + 25309739 T^{10} + 65272284 T^{11} + 612878722 T^{12} + 1049251876 T^{13} - 16289383105 T^{14} - 78416653292 T^{15} - 205836894128 T^{16} - 78416653292 p T^{17} - 16289383105 p^{2} T^{18} + 1049251876 p^{3} T^{19} + 612878722 p^{4} T^{20} + 65272284 p^{5} T^{21} + 25309739 p^{6} T^{22} + 7902460 p^{7} T^{23} + 732467 p^{8} T^{24} - 174480 p^{9} T^{25} - 55264 p^{10} T^{26} - 2984 p^{11} T^{27} - 637 p^{12} T^{28} - 316 p^{13} T^{29} + 7 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 16 T + 57 T^{2} + 192 T^{3} + 68 T^{4} - 9928 T^{5} - 5599 T^{6} - 126432 T^{7} + 3008609 T^{8} - 1908776 T^{9} + 5439712 T^{10} - 593869528 T^{11} + 240477854 T^{12} + 11539277232 T^{13} + 72126154334 T^{14} - 403133954040 T^{15} - 410009080200 T^{16} - 403133954040 p T^{17} + 72126154334 p^{2} T^{18} + 11539277232 p^{3} T^{19} + 240477854 p^{4} T^{20} - 593869528 p^{5} T^{21} + 5439712 p^{6} T^{22} - 1908776 p^{7} T^{23} + 3008609 p^{8} T^{24} - 126432 p^{9} T^{25} - 5599 p^{10} T^{26} - 9928 p^{11} T^{27} + 68 p^{12} T^{28} + 192 p^{13} T^{29} + 57 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 166 T^{2} + 12949 T^{4} - 632166 T^{6} + 22223694 T^{8} - 668560266 T^{10} + 21666695883 T^{12} - 785303223850 T^{14} + 26472704735266 T^{16} - 785303223850 p^{2} T^{18} + 21666695883 p^{4} T^{20} - 668560266 p^{6} T^{22} + 22223694 p^{8} T^{24} - 632166 p^{10} T^{26} + 12949 p^{12} T^{28} - 166 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 + 6 T + 159 T^{2} + 882 T^{3} + 12811 T^{4} + 90240 T^{5} + 791684 T^{6} + 7077372 T^{7} + 42589419 T^{8} + 420195546 T^{9} + 2236955115 T^{10} + 19672611762 T^{11} + 115695284286 T^{12} + 769816568562 T^{13} + 5498435042763 T^{14} + 28033079947506 T^{15} + 223637419109860 T^{16} + 28033079947506 p T^{17} + 5498435042763 p^{2} T^{18} + 769816568562 p^{3} T^{19} + 115695284286 p^{4} T^{20} + 19672611762 p^{5} T^{21} + 2236955115 p^{6} T^{22} + 420195546 p^{7} T^{23} + 42589419 p^{8} T^{24} + 7077372 p^{9} T^{25} + 791684 p^{10} T^{26} + 90240 p^{11} T^{27} + 12811 p^{12} T^{28} + 882 p^{13} T^{29} + 159 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 12 T + 181 T^{2} - 1596 T^{3} + 13092 T^{4} - 83196 T^{5} + 447789 T^{6} - 1535052 T^{7} + 1623065 T^{8} + 53443296 T^{9} - 543147536 T^{10} + 4258142880 T^{11} - 22449074042 T^{12} + 96745145784 T^{13} - 180876262034 T^{14} - 780194868264 T^{15} + 10067773061912 T^{16} - 780194868264 p T^{17} - 180876262034 p^{2} T^{18} + 96745145784 p^{3} T^{19} - 22449074042 p^{4} T^{20} + 4258142880 p^{5} T^{21} - 543147536 p^{6} T^{22} + 53443296 p^{7} T^{23} + 1623065 p^{8} T^{24} - 1535052 p^{9} T^{25} + 447789 p^{10} T^{26} - 83196 p^{11} T^{27} + 13092 p^{12} T^{28} - 1596 p^{13} T^{29} + 181 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 6 T - 189 T^{2} - 1798 T^{3} + 16643 T^{4} + 247148 T^{5} - 740244 T^{6} - 22516656 T^{7} - 9097877 T^{8} + 1515376526 T^{9} + 4773794727 T^{10} - 76461371250 T^{11} - 472577870658 T^{12} + 2704834556962 T^{13} + 30686093801815 T^{14} - 45237468371142 T^{15} - 1498942875101612 T^{16} - 45237468371142 p T^{17} + 30686093801815 p^{2} T^{18} + 2704834556962 p^{3} T^{19} - 472577870658 p^{4} T^{20} - 76461371250 p^{5} T^{21} + 4773794727 p^{6} T^{22} + 1515376526 p^{7} T^{23} - 9097877 p^{8} T^{24} - 22516656 p^{9} T^{25} - 740244 p^{10} T^{26} + 247148 p^{11} T^{27} + 16643 p^{12} T^{28} - 1798 p^{13} T^{29} - 189 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 172 T^{2} + 20310 T^{4} - 1895064 T^{6} + 149501153 T^{8} - 10274305576 T^{10} + 627315503302 T^{12} - 34357180799956 T^{14} + 1698181994903300 T^{16} - 34357180799956 p^{2} T^{18} + 627315503302 p^{4} T^{20} - 10274305576 p^{6} T^{22} + 149501153 p^{8} T^{24} - 1895064 p^{10} T^{26} + 20310 p^{12} T^{28} - 172 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 20 T + 488 T^{2} + 6544 T^{3} + 93562 T^{4} + 949332 T^{5} + 9932896 T^{6} + 79930196 T^{7} + 656774411 T^{8} + 79930196 p T^{9} + 9932896 p^{2} T^{10} + 949332 p^{3} T^{11} + 93562 p^{4} T^{12} + 6544 p^{5} T^{13} + 488 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 6 T + 391 T^{2} - 2274 T^{3} + 82435 T^{4} - 514956 T^{5} + 12300604 T^{6} - 83025336 T^{7} + 1434570239 T^{8} - 10347865326 T^{9} + 138377484031 T^{10} - 1039494650094 T^{11} + 11387928933594 T^{12} - 86355753939450 T^{13} + 815235163087315 T^{14} - 6024366621121554 T^{15} + 51246596450892228 T^{16} - 6024366621121554 p T^{17} + 815235163087315 p^{2} T^{18} - 86355753939450 p^{3} T^{19} + 11387928933594 p^{4} T^{20} - 1039494650094 p^{5} T^{21} + 138377484031 p^{6} T^{22} - 10347865326 p^{7} T^{23} + 1434570239 p^{8} T^{24} - 83025336 p^{9} T^{25} + 12300604 p^{10} T^{26} - 514956 p^{11} T^{27} + 82435 p^{12} T^{28} - 2274 p^{13} T^{29} + 391 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 2 T - 252 T^{2} - 712 T^{3} + 32410 T^{4} + 131190 T^{5} - 2590332 T^{6} - 16985002 T^{7} + 123774253 T^{8} + 1701284262 T^{9} - 831820664 T^{10} - 130113812566 T^{11} - 488444889366 T^{12} + 6954840256176 T^{13} + 57227216725392 T^{14} - 171665844701310 T^{15} - 4134609738368580 T^{16} - 171665844701310 p T^{17} + 57227216725392 p^{2} T^{18} + 6954840256176 p^{3} T^{19} - 488444889366 p^{4} T^{20} - 130113812566 p^{5} T^{21} - 831820664 p^{6} T^{22} + 1701284262 p^{7} T^{23} + 123774253 p^{8} T^{24} - 16985002 p^{9} T^{25} - 2590332 p^{10} T^{26} + 131190 p^{11} T^{27} + 32410 p^{12} T^{28} - 712 p^{13} T^{29} - 252 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 30 T + 735 T^{2} + 13050 T^{3} + 202051 T^{4} + 2702940 T^{5} + 33178988 T^{6} + 376041048 T^{7} + 4042964319 T^{8} + 41472839910 T^{9} + 410196980199 T^{10} + 3931363929078 T^{11} + 36549908746746 T^{12} + 331364719158018 T^{13} + 2916146104868427 T^{14} + 25004214575894442 T^{15} + 207555825988177444 T^{16} + 25004214575894442 p T^{17} + 2916146104868427 p^{2} T^{18} + 331364719158018 p^{3} T^{19} + 36549908746746 p^{4} T^{20} + 3931363929078 p^{5} T^{21} + 410196980199 p^{6} T^{22} + 41472839910 p^{7} T^{23} + 4042964319 p^{8} T^{24} + 376041048 p^{9} T^{25} + 33178988 p^{10} T^{26} + 2702940 p^{11} T^{27} + 202051 p^{12} T^{28} + 13050 p^{13} T^{29} + 735 p^{14} T^{30} + 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 12 T + 187 T^{2} - 1668 T^{3} + 5491 T^{4} - 82776 T^{5} + 569392 T^{6} - 8170224 T^{7} + 122530559 T^{8} - 220685004 T^{9} - 88004789 T^{10} + 19542793140 T^{11} - 448166797530 T^{12} + 216341616348 T^{13} - 19397715309461 T^{14} + 363554627065356 T^{15} - 1768187770175856 T^{16} + 363554627065356 p T^{17} - 19397715309461 p^{2} T^{18} + 216341616348 p^{3} T^{19} - 448166797530 p^{4} T^{20} + 19542793140 p^{5} T^{21} - 88004789 p^{6} T^{22} - 220685004 p^{7} T^{23} + 122530559 p^{8} T^{24} - 8170224 p^{9} T^{25} + 569392 p^{10} T^{26} - 82776 p^{11} T^{27} + 5491 p^{12} T^{28} - 1668 p^{13} T^{29} + 187 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 490 T^{2} + 117949 T^{4} - 18332682 T^{6} + 2035269222 T^{8} - 168157124742 T^{10} + 10521883563027 T^{12} - 532141217479174 T^{14} + 30315689078018386 T^{16} - 532141217479174 p^{2} T^{18} + 10521883563027 p^{4} T^{20} - 168157124742 p^{6} T^{22} + 2035269222 p^{8} T^{24} - 18332682 p^{10} T^{26} + 117949 p^{12} T^{28} - 490 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 8 T + 418 T^{2} + 3044 T^{3} + 86329 T^{4} + 570592 T^{5} + 11477022 T^{6} + 67248340 T^{7} + 1071066508 T^{8} + 67248340 p T^{9} + 11477022 p^{2} T^{10} + 570592 p^{3} T^{11} + 86329 p^{4} T^{12} + 3044 p^{5} T^{13} + 418 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 954 T^{2} + 444045 T^{4} - 134269010 T^{6} + 29564282254 T^{8} - 5028926577566 T^{10} + 683656967390851 T^{12} - 75764207564106646 T^{14} + 6916940745863881634 T^{16} - 75764207564106646 p^{2} T^{18} + 683656967390851 p^{4} T^{20} - 5028926577566 p^{6} T^{22} + 29564282254 p^{8} T^{24} - 134269010 p^{10} T^{26} + 444045 p^{12} T^{28} - 954 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 30 T + 729 T^{2} - 12870 T^{3} + 184020 T^{4} - 2223366 T^{5} + 23277469 T^{6} - 218466390 T^{7} + 2044696885 T^{8} - 19931773944 T^{9} + 212739210064 T^{10} - 2281139666328 T^{11} + 22684899675226 T^{12} - 198460869665820 T^{13} + 1537203323836010 T^{14} - 10973304486885972 T^{15} + 91295276766869624 T^{16} - 10973304486885972 p T^{17} + 1537203323836010 p^{2} T^{18} - 198460869665820 p^{3} T^{19} + 22684899675226 p^{4} T^{20} - 2281139666328 p^{5} T^{21} + 212739210064 p^{6} T^{22} - 19931773944 p^{7} T^{23} + 2044696885 p^{8} T^{24} - 218466390 p^{9} T^{25} + 23277469 p^{10} T^{26} - 2223366 p^{11} T^{27} + 184020 p^{12} T^{28} - 12870 p^{13} T^{29} + 729 p^{14} T^{30} - 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 + 24 T + 560 T^{2} + 8832 T^{3} + 110832 T^{4} + 11472 p T^{5} + 8503424 T^{6} + 51497784 T^{7} + 425262786 T^{8} + 7413688752 T^{9} + 147968588288 T^{10} + 2231375206416 T^{11} + 26291846711104 T^{12} + 236750915178672 T^{13} + 1628120927358640 T^{14} + 8699777160325872 T^{15} + 52186839552165763 T^{16} + 8699777160325872 p T^{17} + 1628120927358640 p^{2} T^{18} + 236750915178672 p^{3} T^{19} + 26291846711104 p^{4} T^{20} + 2231375206416 p^{5} T^{21} + 147968588288 p^{6} T^{22} + 7413688752 p^{7} T^{23} + 425262786 p^{8} T^{24} + 51497784 p^{9} T^{25} + 8503424 p^{10} T^{26} + 11472 p^{12} T^{27} + 110832 p^{12} T^{28} + 8832 p^{13} T^{29} + 560 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.45295352507608754118914341141, −2.39903115219632132371289198084, −2.31384888307251556914184640873, −2.15012273823028261366858999642, −2.04345946472508224100603628453, −1.80557920110899329390622678186, −1.80345447403369989181308352007, −1.78751009987588553996505260975, −1.74211613388906668446811956505, −1.68420540576754858263652638221, −1.67063638579990710772810961720, −1.65395991825429234863099151121, −1.46341374015006688569504604438, −1.37566802431133529661062055954, −1.24957970007362354644070963289, −1.23540456400531634060342546835, −1.15929938600890731764833332438, −0.985118489785762891483811733156, −0.69046869594014072079153430203, −0.68648295231166232757845933070, −0.56020374019418857421822118974, −0.53429645354509012443018569364, −0.53092144055006649249313659005, −0.32632045238722361214638891584, −0.12979598714310611133272292368, 0.12979598714310611133272292368, 0.32632045238722361214638891584, 0.53092144055006649249313659005, 0.53429645354509012443018569364, 0.56020374019418857421822118974, 0.68648295231166232757845933070, 0.69046869594014072079153430203, 0.985118489785762891483811733156, 1.15929938600890731764833332438, 1.23540456400531634060342546835, 1.24957970007362354644070963289, 1.37566802431133529661062055954, 1.46341374015006688569504604438, 1.65395991825429234863099151121, 1.67063638579990710772810961720, 1.68420540576754858263652638221, 1.74211613388906668446811956505, 1.78751009987588553996505260975, 1.80345447403369989181308352007, 1.80557920110899329390622678186, 2.04345946472508224100603628453, 2.15012273823028261366858999642, 2.31384888307251556914184640873, 2.39903115219632132371289198084, 2.45295352507608754118914341141
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.