Dirichlet series
| L(s) = 1 | + 2·2-s + 16·3-s − 2·4-s − 4·5-s + 32·6-s − 4·7-s − 8·8-s + 136·9-s − 8·10-s − 32·12-s − 8·14-s − 64·15-s − 4·16-s − 8·17-s + 272·18-s + 8·19-s + 8·20-s − 64·21-s − 128·24-s + 24·25-s + 816·27-s + 8·28-s + 12·29-s − 128·30-s + 4·32-s − 16·34-s + 16·35-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 9.23·3-s − 4-s − 1.78·5-s + 13.0·6-s − 1.51·7-s − 2.82·8-s + 45.3·9-s − 2.52·10-s − 9.23·12-s − 2.13·14-s − 16.5·15-s − 16-s − 1.94·17-s + 64.1·18-s + 1.83·19-s + 1.78·20-s − 13.9·21-s − 26.1·24-s + 24/5·25-s + 157.·27-s + 1.51·28-s + 2.22·29-s − 23.3·30-s + 0.707·32-s − 2.74·34-s + 2.70·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{16} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(33098.9\) |
| Root analytic conductor: | \(1.38434\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(167.9453119\) |
| \(L(\frac12)\) | \(\approx\) | \(167.9453119\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T + 3 p T^{2} - p^{3} T^{3} + p^{4} T^{4} - 3 p^{2} T^{5} + 5 p^{2} T^{6} + 5 p^{2} T^{8} + 5 p^{4} T^{10} - 3 p^{5} T^{11} + p^{8} T^{12} - p^{8} T^{13} + 3 p^{7} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 3 | \( ( 1 - T )^{16} \) | |
| 5 | \( 1 + 4 T - 8 T^{2} - 44 T^{3} + 32 T^{4} + 188 T^{5} - 232 T^{6} - 68 p T^{7} + 1486 T^{8} - 68 p^{2} T^{9} - 232 p^{2} T^{10} + 188 p^{3} T^{11} + 32 p^{4} T^{12} - 44 p^{5} T^{13} - 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( 1 + 4 T + 8 T^{2} - 4 T^{3} + 172 T^{5} + 696 T^{6} + 2292 T^{7} + 5532 T^{8} + 612 T^{9} - 36184 T^{10} - 53828 T^{11} + 333568 T^{12} + 977996 T^{13} + 340376 T^{14} - 3863884 T^{15} - 12047034 T^{16} - 3863884 p T^{17} + 340376 p^{2} T^{18} + 977996 p^{3} T^{19} + 333568 p^{4} T^{20} - 53828 p^{5} T^{21} - 36184 p^{6} T^{22} + 612 p^{7} T^{23} + 5532 p^{8} T^{24} + 2292 p^{9} T^{25} + 696 p^{10} T^{26} + 172 p^{11} T^{27} - 4 p^{13} T^{29} + 8 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) |
| 11 | \( 1 - 8 T^{3} - 32 T^{4} - 136 T^{5} + 32 T^{6} - 768 T^{7} - 16804 T^{8} - 25600 T^{9} + 16416 T^{10} + 123256 T^{11} - 514016 T^{12} - 565064 T^{13} + 3853504 T^{14} + 32401152 T^{15} + 429795078 T^{16} + 32401152 p T^{17} + 3853504 p^{2} T^{18} - 565064 p^{3} T^{19} - 514016 p^{4} T^{20} + 123256 p^{5} T^{21} + 16416 p^{6} T^{22} - 25600 p^{7} T^{23} - 16804 p^{8} T^{24} - 768 p^{9} T^{25} + 32 p^{10} T^{26} - 136 p^{11} T^{27} - 32 p^{12} T^{28} - 8 p^{13} T^{29} + p^{16} T^{32} \) | |
| 13 | \( 1 - 88 T^{2} + 3936 T^{4} - 121480 T^{6} + 2952732 T^{8} - 60366232 T^{10} + 1069871008 T^{12} - 16663613960 T^{14} + 229854199558 T^{16} - 16663613960 p^{2} T^{18} + 1069871008 p^{4} T^{20} - 60366232 p^{6} T^{22} + 2952732 p^{8} T^{24} - 121480 p^{10} T^{26} + 3936 p^{12} T^{28} - 88 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 + 8 T + 32 T^{2} + 192 T^{3} + 1296 T^{4} + 4832 T^{5} + 15616 T^{6} + 78584 T^{7} + 505468 T^{8} + 2205160 T^{9} + 7990208 T^{10} + 44594272 T^{11} + 256382960 T^{12} + 857331712 T^{13} + 9380128 p^{2} T^{14} + 782258328 p T^{15} + 65658804998 T^{16} + 782258328 p^{2} T^{17} + 9380128 p^{4} T^{18} + 857331712 p^{3} T^{19} + 256382960 p^{4} T^{20} + 44594272 p^{5} T^{21} + 7990208 p^{6} T^{22} + 2205160 p^{7} T^{23} + 505468 p^{8} T^{24} + 78584 p^{9} T^{25} + 15616 p^{10} T^{26} + 4832 p^{11} T^{27} + 1296 p^{12} T^{28} + 192 p^{13} T^{29} + 32 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 8 T + 32 T^{2} - 184 T^{3} + 1192 T^{4} - 7128 T^{5} + 35808 T^{6} - 182440 T^{7} + 968220 T^{8} - 251768 p T^{9} + 23575712 T^{10} - 110017880 T^{11} + 534533400 T^{12} - 2511783672 T^{13} + 11199840608 T^{14} - 50585873416 T^{15} + 220421418630 T^{16} - 50585873416 p T^{17} + 11199840608 p^{2} T^{18} - 2511783672 p^{3} T^{19} + 534533400 p^{4} T^{20} - 110017880 p^{5} T^{21} + 23575712 p^{6} T^{22} - 251768 p^{8} T^{23} + 968220 p^{8} T^{24} - 182440 p^{9} T^{25} + 35808 p^{10} T^{26} - 7128 p^{11} T^{27} + 1192 p^{12} T^{28} - 184 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - 160 T^{3} - 248 T^{4} + 5664 T^{5} + 12800 T^{6} + 107968 T^{7} - 671076 T^{8} - 1509824 T^{9} + 1939968 T^{10} + 42737696 T^{11} + 402709560 T^{12} - 337368736 T^{13} - 490245120 T^{14} + 801086080 T^{15} + 63164018630 T^{16} + 801086080 p T^{17} - 490245120 p^{2} T^{18} - 337368736 p^{3} T^{19} + 402709560 p^{4} T^{20} + 42737696 p^{5} T^{21} + 1939968 p^{6} T^{22} - 1509824 p^{7} T^{23} - 671076 p^{8} T^{24} + 107968 p^{9} T^{25} + 12800 p^{10} T^{26} + 5664 p^{11} T^{27} - 248 p^{12} T^{28} - 160 p^{13} T^{29} + p^{16} T^{32} \) | |
| 29 | \( 1 - 12 T + 72 T^{2} - 4 T^{3} - 656 T^{4} - 8988 T^{5} + 155096 T^{6} - 683732 T^{7} + 877692 T^{8} - 3117356 T^{9} + 119084424 T^{10} - 987113284 T^{11} + 3873688464 T^{12} - 11379902908 T^{13} + 65358101976 T^{14} - 662085368084 T^{15} + 4134892081478 T^{16} - 662085368084 p T^{17} + 65358101976 p^{2} T^{18} - 11379902908 p^{3} T^{19} + 3873688464 p^{4} T^{20} - 987113284 p^{5} T^{21} + 119084424 p^{6} T^{22} - 3117356 p^{7} T^{23} + 877692 p^{8} T^{24} - 683732 p^{9} T^{25} + 155096 p^{10} T^{26} - 8988 p^{11} T^{27} - 656 p^{12} T^{28} - 4 p^{13} T^{29} + 72 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 96 T^{2} + 7112 T^{4} - 388768 T^{6} + 18087708 T^{8} - 721073120 T^{10} + 26350390904 T^{12} - 876800845088 T^{14} + 28138684185670 T^{16} - 876800845088 p^{2} T^{18} + 26350390904 p^{4} T^{20} - 721073120 p^{6} T^{22} + 18087708 p^{8} T^{24} - 388768 p^{10} T^{26} + 7112 p^{12} T^{28} - 96 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 - 328 T^{2} + 52992 T^{4} - 5677272 T^{6} + 456886620 T^{8} - 29517982728 T^{10} + 1590573913088 T^{12} - 73119740727896 T^{14} + 2904354238358022 T^{16} - 73119740727896 p^{2} T^{18} + 1590573913088 p^{4} T^{20} - 29517982728 p^{6} T^{22} + 456886620 p^{8} T^{24} - 5677272 p^{10} T^{26} + 52992 p^{12} T^{28} - 328 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( 1 - 352 T^{2} + 61848 T^{4} - 7239840 T^{6} + 635304668 T^{8} - 44598927328 T^{10} + 2611030729000 T^{12} - 131080145793568 T^{14} + 5738582834880326 T^{16} - 131080145793568 p^{2} T^{18} + 2611030729000 p^{4} T^{20} - 44598927328 p^{6} T^{22} + 635304668 p^{8} T^{24} - 7239840 p^{10} T^{26} + 61848 p^{12} T^{28} - 352 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 256 T^{2} + 32632 T^{4} - 2795520 T^{6} + 186491868 T^{8} - 10606010880 T^{10} + 539127683016 T^{12} - 25153797430528 T^{14} + 1106084382475654 T^{16} - 25153797430528 p^{2} T^{18} + 539127683016 p^{4} T^{20} - 10606010880 p^{6} T^{22} + 186491868 p^{8} T^{24} - 2795520 p^{10} T^{26} + 32632 p^{12} T^{28} - 256 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 + 32 T + 512 T^{2} + 5312 T^{3} + 43176 T^{4} + 353152 T^{5} + 3303424 T^{6} + 29687008 T^{7} + 220234460 T^{8} + 1371720928 T^{9} + 8561945600 T^{10} + 60430684416 T^{11} + 421964783512 T^{12} + 2542806094400 T^{13} + 14218468624896 T^{14} + 87801922468896 T^{15} + 598439585825734 T^{16} + 87801922468896 p T^{17} + 14218468624896 p^{2} T^{18} + 2542806094400 p^{3} T^{19} + 421964783512 p^{4} T^{20} + 60430684416 p^{5} T^{21} + 8561945600 p^{6} T^{22} + 1371720928 p^{7} T^{23} + 220234460 p^{8} T^{24} + 29687008 p^{9} T^{25} + 3303424 p^{10} T^{26} + 353152 p^{11} T^{27} + 43176 p^{12} T^{28} + 5312 p^{13} T^{29} + 512 p^{14} T^{30} + 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 8 T + 312 T^{2} - 2176 T^{3} + 46352 T^{4} - 281744 T^{5} + 4285176 T^{6} - 22453032 T^{7} + 271130734 T^{8} - 22453032 p T^{9} + 4285176 p^{2} T^{10} - 281744 p^{3} T^{11} + 46352 p^{4} T^{12} - 2176 p^{5} T^{13} + 312 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 24 T + 288 T^{2} + 1504 T^{3} - 4768 T^{4} - 109696 T^{5} - 128512 T^{6} + 9859704 T^{7} + 114773468 T^{8} + 507110648 T^{9} - 651247424 T^{10} - 22104746816 T^{11} - 130018266464 T^{12} - 268883047008 T^{13} + 2370020444960 T^{14} + 40501928963672 T^{15} + 414870158287110 T^{16} + 40501928963672 p T^{17} + 2370020444960 p^{2} T^{18} - 268883047008 p^{3} T^{19} - 130018266464 p^{4} T^{20} - 22104746816 p^{5} T^{21} - 651247424 p^{6} T^{22} + 507110648 p^{7} T^{23} + 114773468 p^{8} T^{24} + 9859704 p^{9} T^{25} - 128512 p^{10} T^{26} - 109696 p^{11} T^{27} - 4768 p^{12} T^{28} + 1504 p^{13} T^{29} + 288 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 40 T + 800 T^{2} - 10840 T^{3} + 109080 T^{4} - 792968 T^{5} + 3207520 T^{6} + 12113864 T^{7} - 373671012 T^{8} + 3954939256 T^{9} - 26052203488 T^{10} + 91316522376 T^{11} + 190100482600 T^{12} - 4912653370088 T^{13} + 32118825602400 T^{14} - 87618367306328 T^{15} + 22971298530438 T^{16} - 87618367306328 p T^{17} + 32118825602400 p^{2} T^{18} - 4912653370088 p^{3} T^{19} + 190100482600 p^{4} T^{20} + 91316522376 p^{5} T^{21} - 26052203488 p^{6} T^{22} + 3954939256 p^{7} T^{23} - 373671012 p^{8} T^{24} + 12113864 p^{9} T^{25} + 3207520 p^{10} T^{26} - 792968 p^{11} T^{27} + 109080 p^{12} T^{28} - 10840 p^{13} T^{29} + 800 p^{14} T^{30} - 40 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 832 T^{2} + 334680 T^{4} - 86524480 T^{6} + 16108437276 T^{8} - 2291695812544 T^{10} + 257849839451368 T^{12} - 23412006337579712 T^{14} + 1733902880672576902 T^{16} - 23412006337579712 p^{2} T^{18} + 257849839451368 p^{4} T^{20} - 2291695812544 p^{6} T^{22} + 16108437276 p^{8} T^{24} - 86524480 p^{10} T^{26} + 334680 p^{12} T^{28} - 832 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 176 T^{2} + 112 T^{3} + 16796 T^{4} + 19344 T^{5} + 1677520 T^{6} + 1371808 T^{7} + 147932550 T^{8} + 1371808 p T^{9} + 1677520 p^{2} T^{10} + 19344 p^{3} T^{11} + 16796 p^{4} T^{12} + 112 p^{5} T^{13} + 176 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 - 8 T + 32 T^{2} - 376 T^{3} + 10776 T^{4} - 171656 T^{5} + 1099104 T^{6} - 12208248 T^{7} + 145630428 T^{8} - 1322639016 T^{9} + 13400186144 T^{10} - 122947983320 T^{11} + 1461979003048 T^{12} - 11241533149096 T^{13} + 86473569544544 T^{14} - 944871083629144 T^{15} + 8169395568969030 T^{16} - 944871083629144 p T^{17} + 86473569544544 p^{2} T^{18} - 11241533149096 p^{3} T^{19} + 1461979003048 p^{4} T^{20} - 122947983320 p^{5} T^{21} + 13400186144 p^{6} T^{22} - 1322639016 p^{7} T^{23} + 145630428 p^{8} T^{24} - 12208248 p^{9} T^{25} + 1099104 p^{10} T^{26} - 171656 p^{11} T^{27} + 10776 p^{12} T^{28} - 376 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 24 T + 688 T^{2} + 10760 T^{3} + 182340 T^{4} + 2163048 T^{5} + 27231888 T^{6} + 260060088 T^{7} + 2625684550 T^{8} + 260060088 p T^{9} + 27231888 p^{2} T^{10} + 2163048 p^{3} T^{11} + 182340 p^{4} T^{12} + 10760 p^{5} T^{13} + 688 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 4 T + 440 T^{2} + 1060 T^{3} + 92508 T^{4} + 140452 T^{5} + 12703176 T^{6} + 14471140 T^{7} + 1242402022 T^{8} + 14471140 p T^{9} + 12703176 p^{2} T^{10} + 140452 p^{3} T^{11} + 92508 p^{4} T^{12} + 1060 p^{5} T^{13} + 440 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 368 T^{2} + 16 p T^{3} + 63260 T^{4} + 490128 T^{5} + 7260688 T^{6} + 77256800 T^{7} + 679711110 T^{8} + 77256800 p T^{9} + 7260688 p^{2} T^{10} + 490128 p^{3} T^{11} + 63260 p^{4} T^{12} + 16 p^{6} T^{13} + 368 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 48 T + 1152 T^{2} - 18032 T^{3} + 197848 T^{4} - 1611600 T^{5} + 12012416 T^{6} - 1216912 p T^{7} + 1320763356 T^{8} - 11021148208 T^{9} + 57996173952 T^{10} - 481959652208 T^{11} + 12659819092200 T^{12} - 235613637176016 T^{13} + 2804511567646080 T^{14} - 25136893111378960 T^{15} + 223711349593072326 T^{16} - 25136893111378960 p T^{17} + 2804511567646080 p^{2} T^{18} - 235613637176016 p^{3} T^{19} + 12659819092200 p^{4} T^{20} - 481959652208 p^{5} T^{21} + 57996173952 p^{6} T^{22} - 11021148208 p^{7} T^{23} + 1320763356 p^{8} T^{24} - 1216912 p^{10} T^{25} + 12012416 p^{10} T^{26} - 1611600 p^{11} T^{27} + 197848 p^{12} T^{28} - 18032 p^{13} T^{29} + 1152 p^{14} T^{30} - 48 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−3.52980441195405276640340882485, −3.34014145630113611952698583397, −3.14393138082321034213637388165, −3.09946216358576318400544730714, −2.99745562218334358860100560501, −2.97267014213894544179917428637, −2.96223295566897583121723761119, −2.91738582877832170663101134183, −2.89331882967798078812821476149, −2.84799349191350100892605020185, −2.83216136779539023217383874721, −2.65463885879279742873597854966, −2.58285792517752135436710432922, −2.33153715389657729227335330241, −2.25073116764673599137123261887, −2.18450486600889985585508527743, −1.90385095972494446443205662927, −1.76021528153817121964946328420, −1.69500213047663931395373284405, −1.68384408308562354372505558581, −1.56271863038314469110503647982, −1.34121579305514309465991624559, −1.02561186069549654676554459809, −0.951475304602595604941625985241, −0.58175488167602208106421502868, 0.58175488167602208106421502868, 0.951475304602595604941625985241, 1.02561186069549654676554459809, 1.34121579305514309465991624559, 1.56271863038314469110503647982, 1.68384408308562354372505558581, 1.69500213047663931395373284405, 1.76021528153817121964946328420, 1.90385095972494446443205662927, 2.18450486600889985585508527743, 2.25073116764673599137123261887, 2.33153715389657729227335330241, 2.58285792517752135436710432922, 2.65463885879279742873597854966, 2.83216136779539023217383874721, 2.84799349191350100892605020185, 2.89331882967798078812821476149, 2.91738582877832170663101134183, 2.96223295566897583121723761119, 2.97267014213894544179917428637, 2.99745562218334358860100560501, 3.09946216358576318400544730714, 3.14393138082321034213637388165, 3.34014145630113611952698583397, 3.52980441195405276640340882485
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.