Dirichlet series
| L(s) = 1 | + 6·11-s + 3·13-s + 9·17-s − 21·23-s + 16·25-s − 6·29-s + 37-s − 6·41-s − 2·43-s − 36·47-s − 30·59-s + 14·67-s + 2·79-s + 21·89-s + 3·97-s + 24·101-s + 21·103-s + 87·107-s − 14·109-s + 9·113-s − 25·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 18·143-s + ⋯ |
| L(s) = 1 | + 1.80·11-s + 0.832·13-s + 2.18·17-s − 4.37·23-s + 16/5·25-s − 1.11·29-s + 0.164·37-s − 0.937·41-s − 0.304·43-s − 5.25·47-s − 3.90·59-s + 1.71·67-s + 0.225·79-s + 2.22·89-s + 0.304·97-s + 2.38·101-s + 2.06·103-s + 8.41·107-s − 1.34·109-s + 0.846·113-s − 2.27·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.50·143-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{48} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.03359\times 10^{26}\) |
| Root analytic conductor: | \(6.50052\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{48} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(47.65592444\) |
| \(L(\frac12)\) | \(\approx\) | \(47.65592444\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - 16 T^{2} - 24 T^{3} + 21 p T^{4} + 357 T^{5} - 16 p T^{6} - 2232 T^{7} - 2857 T^{8} + 6027 T^{9} + 16041 T^{10} + 777 T^{11} - 31496 T^{12} - 32724 T^{13} + 43664 T^{14} + 15327 T^{15} - 229716 T^{16} + 15327 p T^{17} + 43664 p^{2} T^{18} - 32724 p^{3} T^{19} - 31496 p^{4} T^{20} + 777 p^{5} T^{21} + 16041 p^{6} T^{22} + 6027 p^{7} T^{23} - 2857 p^{8} T^{24} - 2232 p^{9} T^{25} - 16 p^{11} T^{26} + 357 p^{11} T^{27} + 21 p^{13} T^{28} - 24 p^{13} T^{29} - 16 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 - 6 T + 61 T^{2} - 294 T^{3} + 1701 T^{4} - 564 p T^{5} + 24845 T^{6} - 59553 T^{7} + 143666 T^{8} + 78393 T^{9} - 1097631 T^{10} + 8096325 T^{11} - 21925205 T^{12} + 68059401 T^{13} - 34454504 T^{14} - 43343730 T^{15} + 1352446632 T^{16} - 43343730 p T^{17} - 34454504 p^{2} T^{18} + 68059401 p^{3} T^{19} - 21925205 p^{4} T^{20} + 8096325 p^{5} T^{21} - 1097631 p^{6} T^{22} + 78393 p^{7} T^{23} + 143666 p^{8} T^{24} - 59553 p^{9} T^{25} + 24845 p^{10} T^{26} - 564 p^{12} T^{27} + 1701 p^{12} T^{28} - 294 p^{13} T^{29} + 61 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 3 T + 53 T^{2} - 150 T^{3} + 1317 T^{4} - 4452 T^{5} + 23056 T^{6} - 90522 T^{7} + 327950 T^{8} - 1218621 T^{9} + 4082037 T^{10} - 12551466 T^{11} + 42977824 T^{12} - 136810599 T^{13} + 317458436 T^{14} - 1828010007 T^{15} + 2359677564 T^{16} - 1828010007 p T^{17} + 317458436 p^{2} T^{18} - 136810599 p^{3} T^{19} + 42977824 p^{4} T^{20} - 12551466 p^{5} T^{21} + 4082037 p^{6} T^{22} - 1218621 p^{7} T^{23} + 327950 p^{8} T^{24} - 90522 p^{9} T^{25} + 23056 p^{10} T^{26} - 4452 p^{11} T^{27} + 1317 p^{12} T^{28} - 150 p^{13} T^{29} + 53 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 9 T - 31 T^{2} + 498 T^{3} + 6 T^{4} - 11613 T^{5} + 1180 T^{6} + 197118 T^{7} + 356723 T^{8} - 3738756 T^{9} - 13114257 T^{10} + 60977685 T^{11} + 321880693 T^{12} - 548795961 T^{13} - 7936687522 T^{14} + 1772823582 T^{15} + 160937528238 T^{16} + 1772823582 p T^{17} - 7936687522 p^{2} T^{18} - 548795961 p^{3} T^{19} + 321880693 p^{4} T^{20} + 60977685 p^{5} T^{21} - 13114257 p^{6} T^{22} - 3738756 p^{7} T^{23} + 356723 p^{8} T^{24} + 197118 p^{9} T^{25} + 1180 p^{10} T^{26} - 11613 p^{11} T^{27} + 6 p^{12} T^{28} + 498 p^{13} T^{29} - 31 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 59 T^{2} + 78 p T^{4} - 279 T^{5} + 21358 T^{6} + 23166 T^{7} + 284075 T^{8} + 1924515 T^{9} + 8259087 T^{10} + 60795468 T^{11} + 210571285 T^{12} + 1125692802 T^{13} + 3051652292 T^{14} + 14070928500 T^{15} + 40349090466 T^{16} + 14070928500 p T^{17} + 3051652292 p^{2} T^{18} + 1125692802 p^{3} T^{19} + 210571285 p^{4} T^{20} + 60795468 p^{5} T^{21} + 8259087 p^{6} T^{22} + 1924515 p^{7} T^{23} + 284075 p^{8} T^{24} + 23166 p^{9} T^{25} + 21358 p^{10} T^{26} - 279 p^{11} T^{27} + 78 p^{13} T^{28} + 59 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 21 T + 283 T^{2} + 2856 T^{3} + 23904 T^{4} + 177009 T^{5} + 1214978 T^{6} + 7994925 T^{7} + 50784536 T^{8} + 310700127 T^{9} + 1817756595 T^{10} + 10184685783 T^{11} + 55106115085 T^{12} + 290568927984 T^{13} + 1499561670325 T^{14} + 7546165977213 T^{15} + 36821082385998 T^{16} + 7546165977213 p T^{17} + 1499561670325 p^{2} T^{18} + 290568927984 p^{3} T^{19} + 55106115085 p^{4} T^{20} + 10184685783 p^{5} T^{21} + 1817756595 p^{6} T^{22} + 310700127 p^{7} T^{23} + 50784536 p^{8} T^{24} + 7994925 p^{9} T^{25} + 1214978 p^{10} T^{26} + 177009 p^{11} T^{27} + 23904 p^{12} T^{28} + 2856 p^{13} T^{29} + 283 p^{14} T^{30} + 21 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 6 T + 106 T^{2} + 564 T^{3} + 5445 T^{4} + 23700 T^{5} + 153455 T^{6} + 499140 T^{7} + 2268674 T^{8} + 3839766 T^{9} + 25367247 T^{10} + 108756381 T^{11} + 2283543946 T^{12} + 15452150286 T^{13} + 152754700246 T^{14} + 806878620189 T^{15} + 5726182499856 T^{16} + 806878620189 p T^{17} + 152754700246 p^{2} T^{18} + 15452150286 p^{3} T^{19} + 2283543946 p^{4} T^{20} + 108756381 p^{5} T^{21} + 25367247 p^{6} T^{22} + 3839766 p^{7} T^{23} + 2268674 p^{8} T^{24} + 499140 p^{9} T^{25} + 153455 p^{10} T^{26} + 23700 p^{11} T^{27} + 5445 p^{12} T^{28} + 564 p^{13} T^{29} + 106 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 220 T^{2} + 25434 T^{4} - 2035931 T^{6} + 125854031 T^{8} - 6377263569 T^{10} + 274749236767 T^{12} - 10292358079372 T^{14} + 339402697885782 T^{16} - 10292358079372 p^{2} T^{18} + 274749236767 p^{4} T^{20} - 6377263569 p^{6} T^{22} + 125854031 p^{8} T^{24} - 2035931 p^{10} T^{26} + 25434 p^{12} T^{28} - 220 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 - T - 188 T^{2} + 55 T^{3} + 18431 T^{4} + 6958 T^{5} - 1220598 T^{6} - 1466820 T^{7} + 61023285 T^{8} + 128763363 T^{9} - 65590950 p T^{10} - 7087005807 T^{11} + 80710069782 T^{12} + 247085609469 T^{13} - 2444398029834 T^{14} - 3801878390997 T^{15} + 81128049964254 T^{16} - 3801878390997 p T^{17} - 2444398029834 p^{2} T^{18} + 247085609469 p^{3} T^{19} + 80710069782 p^{4} T^{20} - 7087005807 p^{5} T^{21} - 65590950 p^{7} T^{22} + 128763363 p^{7} T^{23} + 61023285 p^{8} T^{24} - 1466820 p^{9} T^{25} - 1220598 p^{10} T^{26} + 6958 p^{11} T^{27} + 18431 p^{12} T^{28} + 55 p^{13} T^{29} - 188 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 6 T - 142 T^{2} - 1146 T^{3} + 7575 T^{4} + 89148 T^{5} - 186515 T^{6} - 3858942 T^{7} + 5165546 T^{8} + 137269398 T^{9} - 383379603 T^{10} - 6275799867 T^{11} + 14048576428 T^{12} + 262470296220 T^{13} + 63020850350 T^{14} - 4818424458177 T^{15} - 18114488698896 T^{16} - 4818424458177 p T^{17} + 63020850350 p^{2} T^{18} + 262470296220 p^{3} T^{19} + 14048576428 p^{4} T^{20} - 6275799867 p^{5} T^{21} - 383379603 p^{6} T^{22} + 137269398 p^{7} T^{23} + 5165546 p^{8} T^{24} - 3858942 p^{9} T^{25} - 186515 p^{10} T^{26} + 89148 p^{11} T^{27} + 7575 p^{12} T^{28} - 1146 p^{13} T^{29} - 142 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 2 T - 137 T^{2} - 620 T^{3} + 8660 T^{4} + 62377 T^{5} - 131688 T^{6} - 3324402 T^{7} - 16902063 T^{8} + 47132751 T^{9} + 1398120027 T^{10} + 5063156292 T^{11} - 36164979717 T^{12} - 388120390392 T^{13} - 815732360646 T^{14} + 8046953157708 T^{15} + 93229603031718 T^{16} + 8046953157708 p T^{17} - 815732360646 p^{2} T^{18} - 388120390392 p^{3} T^{19} - 36164979717 p^{4} T^{20} + 5063156292 p^{5} T^{21} + 1398120027 p^{6} T^{22} + 47132751 p^{7} T^{23} - 16902063 p^{8} T^{24} - 3324402 p^{9} T^{25} - 131688 p^{10} T^{26} + 62377 p^{11} T^{27} + 8660 p^{12} T^{28} - 620 p^{13} T^{29} - 137 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 18 T + 262 T^{2} + 2454 T^{3} + 20893 T^{4} + 154293 T^{5} + 26963 p T^{6} + 9657933 T^{7} + 72865294 T^{8} + 9657933 p T^{9} + 26963 p^{3} T^{10} + 154293 p^{3} T^{11} + 20893 p^{4} T^{12} + 2454 p^{5} T^{13} + 262 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 + 271 T^{2} + 39798 T^{4} - 8019 T^{5} + 3882470 T^{6} - 3112830 T^{7} + 272618249 T^{8} - 573793713 T^{9} + 13751201223 T^{10} - 69938411256 T^{11} + 461723008303 T^{12} - 6218775316710 T^{13} + 7202514917530 T^{14} - 423085276358208 T^{15} - 24546838659498 T^{16} - 423085276358208 p T^{17} + 7202514917530 p^{2} T^{18} - 6218775316710 p^{3} T^{19} + 461723008303 p^{4} T^{20} - 69938411256 p^{5} T^{21} + 13751201223 p^{6} T^{22} - 573793713 p^{7} T^{23} + 272618249 p^{8} T^{24} - 3112830 p^{9} T^{25} + 3882470 p^{10} T^{26} - 8019 p^{11} T^{27} + 39798 p^{12} T^{28} + 271 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 15 T + 304 T^{2} + 2994 T^{3} + 35017 T^{4} + 245181 T^{5} + 2158909 T^{6} + 12195126 T^{7} + 111191710 T^{8} + 12195126 p T^{9} + 2158909 p^{2} T^{10} + 245181 p^{3} T^{11} + 35017 p^{4} T^{12} + 2994 p^{5} T^{13} + 304 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 745 T^{2} + 270666 T^{4} - 63553928 T^{6} + 10780728317 T^{8} - 1399459132305 T^{10} + 143698232661445 T^{12} - 11899376361024010 T^{14} + 802850906302349022 T^{16} - 11899376361024010 p^{2} T^{18} + 143698232661445 p^{4} T^{20} - 1399459132305 p^{6} T^{22} + 10780728317 p^{8} T^{24} - 63553928 p^{10} T^{26} + 270666 p^{12} T^{28} - 745 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 7 T + 315 T^{2} - 1580 T^{3} + 47600 T^{4} - 182439 T^{5} + 4735936 T^{6} - 14472934 T^{7} + 354986784 T^{8} - 14472934 p T^{9} + 4735936 p^{2} T^{10} - 182439 p^{3} T^{11} + 47600 p^{4} T^{12} - 1580 p^{5} T^{13} + 315 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 389 T^{2} + 95718 T^{4} - 239186 p T^{6} + 2417461733 T^{8} - 285171142929 T^{10} + 28739565041956 T^{12} - 2501948752776044 T^{14} + 190114711967546784 T^{16} - 2501948752776044 p^{2} T^{18} + 28739565041956 p^{4} T^{20} - 285171142929 p^{6} T^{22} + 2417461733 p^{8} T^{24} - 239186 p^{11} T^{26} + 95718 p^{12} T^{28} - 389 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 + 278 T^{2} + 39699 T^{4} + 160362 T^{5} + 4049527 T^{6} + 43185312 T^{7} + 316493258 T^{8} + 5978310966 T^{9} + 30516141699 T^{10} + 573490190745 T^{11} + 3892519182646 T^{12} + 40210747978752 T^{13} + 436575836052170 T^{14} + 2488085380264887 T^{15} + 37348173312201540 T^{16} + 2488085380264887 p T^{17} + 436575836052170 p^{2} T^{18} + 40210747978752 p^{3} T^{19} + 3892519182646 p^{4} T^{20} + 573490190745 p^{5} T^{21} + 30516141699 p^{6} T^{22} + 5978310966 p^{7} T^{23} + 316493258 p^{8} T^{24} + 43185312 p^{9} T^{25} + 4049527 p^{10} T^{26} + 160362 p^{11} T^{27} + 39699 p^{12} T^{28} + 278 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - T + 489 T^{2} - 92 T^{3} + 108962 T^{4} + 44511 T^{5} + 14856838 T^{6} + 9973310 T^{7} + 1391605452 T^{8} + 9973310 p T^{9} + 14856838 p^{2} T^{10} + 44511 p^{3} T^{11} + 108962 p^{4} T^{12} - 92 p^{5} T^{13} + 489 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 334 T^{2} - 1272 T^{3} + 54816 T^{4} + 395796 T^{5} - 5397056 T^{6} - 60133572 T^{7} + 310945691 T^{8} + 6455120304 T^{9} - 3389068248 T^{10} - 563598757704 T^{11} - 2368219763012 T^{12} + 39202056035964 T^{13} + 437295746267318 T^{14} - 1352773893773628 T^{15} - 45259898774669136 T^{16} - 1352773893773628 p T^{17} + 437295746267318 p^{2} T^{18} + 39202056035964 p^{3} T^{19} - 2368219763012 p^{4} T^{20} - 563598757704 p^{5} T^{21} - 3389068248 p^{6} T^{22} + 6455120304 p^{7} T^{23} + 310945691 p^{8} T^{24} - 60133572 p^{9} T^{25} - 5397056 p^{10} T^{26} + 395796 p^{11} T^{27} + 54816 p^{12} T^{28} - 1272 p^{13} T^{29} - 334 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 21 T - 253 T^{2} + 6084 T^{3} + 64134 T^{4} - 1086543 T^{5} - 14444834 T^{6} + 154075794 T^{7} + 2457441293 T^{8} - 16916386392 T^{9} - 343956627405 T^{10} + 1437006555873 T^{11} + 41653185355711 T^{12} - 92739461592729 T^{13} - 4399434577615834 T^{14} + 3097874885577318 T^{15} + 411982470004053510 T^{16} + 3097874885577318 p T^{17} - 4399434577615834 p^{2} T^{18} - 92739461592729 p^{3} T^{19} + 41653185355711 p^{4} T^{20} + 1437006555873 p^{5} T^{21} - 343956627405 p^{6} T^{22} - 16916386392 p^{7} T^{23} + 2457441293 p^{8} T^{24} + 154075794 p^{9} T^{25} - 14444834 p^{10} T^{26} - 1086543 p^{11} T^{27} + 64134 p^{12} T^{28} + 6084 p^{13} T^{29} - 253 p^{14} T^{30} - 21 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 3 T + 392 T^{2} - 1167 T^{3} + 75243 T^{4} - 248802 T^{5} + 9281977 T^{6} - 31870980 T^{7} + 787182476 T^{8} - 2086019346 T^{9} + 40966656945 T^{10} + 99548858823 T^{11} - 1131878124686 T^{12} + 48785193558327 T^{13} - 618766130152816 T^{14} + 7413615555033420 T^{15} - 81803373889459032 T^{16} + 7413615555033420 p T^{17} - 618766130152816 p^{2} T^{18} + 48785193558327 p^{3} T^{19} - 1131878124686 p^{4} T^{20} + 99548858823 p^{5} T^{21} + 40966656945 p^{6} T^{22} - 2086019346 p^{7} T^{23} + 787182476 p^{8} T^{24} - 31870980 p^{9} T^{25} + 9281977 p^{10} T^{26} - 248802 p^{11} T^{27} + 75243 p^{12} T^{28} - 1167 p^{13} T^{29} + 392 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.02051361837453416934749806933, −1.83402895424810840473668201931, −1.73139684338361237408258065947, −1.64383124950260564483747049732, −1.58288947316968802854207723469, −1.57246273198988950742622333312, −1.56211943514012981242934453640, −1.46045813170748020779091613306, −1.40395138314900814502393107286, −1.39363952627593428361606926497, −1.29080010433101174981198718958, −1.26271899748059485440825500715, −1.23354394312995586812898301245, −1.13155235150422696965067720946, −0.953669073226929319944918935376, −0.909602362976145479831664344278, −0.74976603137109482935398262669, −0.64161528420349713396607483210, −0.60111878161945676788049361807, −0.50165949474639817383062479177, −0.47759105023215146941205574218, −0.47164921621660950351727199903, −0.27583486044227342634962154120, −0.21093117811083473456909783328, −0.15566694410708169570411721236, 0.15566694410708169570411721236, 0.21093117811083473456909783328, 0.27583486044227342634962154120, 0.47164921621660950351727199903, 0.47759105023215146941205574218, 0.50165949474639817383062479177, 0.60111878161945676788049361807, 0.64161528420349713396607483210, 0.74976603137109482935398262669, 0.909602362976145479831664344278, 0.953669073226929319944918935376, 1.13155235150422696965067720946, 1.23354394312995586812898301245, 1.26271899748059485440825500715, 1.29080010433101174981198718958, 1.39363952627593428361606926497, 1.40395138314900814502393107286, 1.46045813170748020779091613306, 1.56211943514012981242934453640, 1.57246273198988950742622333312, 1.58288947316968802854207723469, 1.64383124950260564483747049732, 1.73139684338361237408258065947, 1.83402895424810840473668201931, 2.02051361837453416934749806933
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.