Dirichlet series
| L(s) = 1 | − 3·2-s − 12·5-s + 7·7-s + 10·8-s + 36·10-s − 4·11-s + 13·13-s − 21·14-s − 14·16-s + 4·17-s + 14·19-s + 12·22-s − 11·23-s + 78·25-s − 39·26-s + 7·29-s + 4·31-s + 32-s − 12·34-s − 84·35-s + 24·37-s − 42·38-s − 120·40-s − 13·41-s + 25·43-s + 33·46-s − 19·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 5.36·5-s + 2.64·7-s + 3.53·8-s + 11.3·10-s − 1.20·11-s + 3.60·13-s − 5.61·14-s − 7/2·16-s + 0.970·17-s + 3.21·19-s + 2.55·22-s − 2.29·23-s + 78/5·25-s − 7.64·26-s + 1.29·29-s + 0.718·31-s + 0.176·32-s − 2.05·34-s − 14.1·35-s + 3.94·37-s − 6.81·38-s − 18.9·40-s − 2.03·41-s + 3.81·43-s + 4.86·46-s − 2.77·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 107^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 107^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{12} \cdot 107^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.04347\times 10^{19}\) |
| Root analytic conductor: | \(6.20064\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{12} \cdot 107^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.367356148\) |
| \(L(\frac12)\) | \(\approx\) | \(2.367356148\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{12} \) | |
| 107 | \( ( 1 + T )^{12} \) | |
| good | 2 | \( 1 + 3 T + 9 T^{2} + 17 T^{3} + 35 T^{4} + 7 p^{3} T^{5} + 13 p^{3} T^{6} + 147 T^{7} + 241 T^{8} + 303 T^{9} + 483 T^{10} + 607 T^{11} + 495 p T^{12} + 607 p T^{13} + 483 p^{2} T^{14} + 303 p^{3} T^{15} + 241 p^{4} T^{16} + 147 p^{5} T^{17} + 13 p^{9} T^{18} + 7 p^{10} T^{19} + 35 p^{8} T^{20} + 17 p^{9} T^{21} + 9 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - p T + 62 T^{2} - 311 T^{3} + 1670 T^{4} - 6459 T^{5} + 26102 T^{6} - 11707 p T^{7} + 272366 T^{8} - 727003 T^{9} + 307691 p T^{10} - 5269232 T^{11} + 15199704 T^{12} - 5269232 p T^{13} + 307691 p^{3} T^{14} - 727003 p^{3} T^{15} + 272366 p^{4} T^{16} - 11707 p^{6} T^{17} + 26102 p^{6} T^{18} - 6459 p^{7} T^{19} + 1670 p^{8} T^{20} - 311 p^{9} T^{21} + 62 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 4 T + 71 T^{2} + 225 T^{3} + 2507 T^{4} + 6884 T^{5} + 60373 T^{6} + 147811 T^{7} + 1106593 T^{8} + 2450548 T^{9} + 16270225 T^{10} + 32735028 T^{11} + 196536584 T^{12} + 32735028 p T^{13} + 16270225 p^{2} T^{14} + 2450548 p^{3} T^{15} + 1106593 p^{4} T^{16} + 147811 p^{5} T^{17} + 60373 p^{6} T^{18} + 6884 p^{7} T^{19} + 2507 p^{8} T^{20} + 225 p^{9} T^{21} + 71 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - p T + 158 T^{2} - 1253 T^{3} + 8955 T^{4} - 50533 T^{5} + 257550 T^{6} - 1084533 T^{7} + 4143643 T^{8} - 13099282 T^{9} + 39103364 T^{10} - 102585378 T^{11} + 343959826 T^{12} - 102585378 p T^{13} + 39103364 p^{2} T^{14} - 13099282 p^{3} T^{15} + 4143643 p^{4} T^{16} - 1084533 p^{5} T^{17} + 257550 p^{6} T^{18} - 50533 p^{7} T^{19} + 8955 p^{8} T^{20} - 1253 p^{9} T^{21} + 158 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 4 T + 126 T^{2} - 474 T^{3} + 7914 T^{4} - 27556 T^{5} + 328250 T^{6} - 1048990 T^{7} + 10033492 T^{8} - 29272838 T^{9} + 238459635 T^{10} - 630910608 T^{11} + 4521226680 T^{12} - 630910608 p T^{13} + 238459635 p^{2} T^{14} - 29272838 p^{3} T^{15} + 10033492 p^{4} T^{16} - 1048990 p^{5} T^{17} + 328250 p^{6} T^{18} - 27556 p^{7} T^{19} + 7914 p^{8} T^{20} - 474 p^{9} T^{21} + 126 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 14 T + 222 T^{2} - 2176 T^{3} + 20968 T^{4} - 161233 T^{5} + 1184744 T^{6} - 7567216 T^{7} + 45931914 T^{8} - 251492114 T^{9} + 1308721771 T^{10} - 6246635028 T^{11} + 28361055576 T^{12} - 6246635028 p T^{13} + 1308721771 p^{2} T^{14} - 251492114 p^{3} T^{15} + 45931914 p^{4} T^{16} - 7567216 p^{5} T^{17} + 1184744 p^{6} T^{18} - 161233 p^{7} T^{19} + 20968 p^{8} T^{20} - 2176 p^{9} T^{21} + 222 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 11 T + 178 T^{2} + 1364 T^{3} + 12608 T^{4} + 74337 T^{5} + 502410 T^{6} + 2480470 T^{7} + 13932815 T^{8} + 64351423 T^{9} + 332804068 T^{10} + 1551872265 T^{11} + 7732345888 T^{12} + 1551872265 p T^{13} + 332804068 p^{2} T^{14} + 64351423 p^{3} T^{15} + 13932815 p^{4} T^{16} + 2480470 p^{5} T^{17} + 502410 p^{6} T^{18} + 74337 p^{7} T^{19} + 12608 p^{8} T^{20} + 1364 p^{9} T^{21} + 178 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 7 T + 238 T^{2} - 1243 T^{3} + 874 p T^{4} - 100106 T^{5} + 1659740 T^{6} - 4890787 T^{7} + 77218550 T^{8} - 167498073 T^{9} + 2819727445 T^{10} - 4770412802 T^{11} + 87344460556 T^{12} - 4770412802 p T^{13} + 2819727445 p^{2} T^{14} - 167498073 p^{3} T^{15} + 77218550 p^{4} T^{16} - 4890787 p^{5} T^{17} + 1659740 p^{6} T^{18} - 100106 p^{7} T^{19} + 874 p^{9} T^{20} - 1243 p^{9} T^{21} + 238 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 4 T + 216 T^{2} - 891 T^{3} + 23071 T^{4} - 3107 p T^{5} + 1615192 T^{6} - 6735368 T^{7} + 83443547 T^{8} - 342115836 T^{9} + 3412761360 T^{10} - 13394847322 T^{11} + 115429793162 T^{12} - 13394847322 p T^{13} + 3412761360 p^{2} T^{14} - 342115836 p^{3} T^{15} + 83443547 p^{4} T^{16} - 6735368 p^{5} T^{17} + 1615192 p^{6} T^{18} - 3107 p^{8} T^{19} + 23071 p^{8} T^{20} - 891 p^{9} T^{21} + 216 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 24 T + 402 T^{2} - 4700 T^{3} + 47007 T^{4} - 401200 T^{5} + 3201466 T^{6} - 23376228 T^{7} + 164091835 T^{8} - 1080494884 T^{9} + 7008397092 T^{10} - 43616671108 T^{11} + 270649660298 T^{12} - 43616671108 p T^{13} + 7008397092 p^{2} T^{14} - 1080494884 p^{3} T^{15} + 164091835 p^{4} T^{16} - 23376228 p^{5} T^{17} + 3201466 p^{6} T^{18} - 401200 p^{7} T^{19} + 47007 p^{8} T^{20} - 4700 p^{9} T^{21} + 402 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 13 T + 322 T^{2} + 2879 T^{3} + 41842 T^{4} + 273200 T^{5} + 3067086 T^{6} + 14334747 T^{7} + 146169460 T^{8} + 426645477 T^{9} + 5148557205 T^{10} + 7154495096 T^{11} + 182171425072 T^{12} + 7154495096 p T^{13} + 5148557205 p^{2} T^{14} + 426645477 p^{3} T^{15} + 146169460 p^{4} T^{16} + 14334747 p^{5} T^{17} + 3067086 p^{6} T^{18} + 273200 p^{7} T^{19} + 41842 p^{8} T^{20} + 2879 p^{9} T^{21} + 322 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 25 T + 545 T^{2} - 8245 T^{3} + 113842 T^{4} - 1306425 T^{5} + 13981882 T^{6} - 132068479 T^{7} + 1179662253 T^{8} - 9564050379 T^{9} + 73973492745 T^{10} - 526500702508 T^{11} + 3591384579552 T^{12} - 526500702508 p T^{13} + 73973492745 p^{2} T^{14} - 9564050379 p^{3} T^{15} + 1179662253 p^{4} T^{16} - 132068479 p^{5} T^{17} + 13981882 p^{6} T^{18} - 1306425 p^{7} T^{19} + 113842 p^{8} T^{20} - 8245 p^{9} T^{21} + 545 p^{10} T^{22} - 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 19 T + 408 T^{2} + 5689 T^{3} + 80509 T^{4} + 909849 T^{5} + 10162416 T^{6} + 97798751 T^{7} + 920725379 T^{8} + 7718223732 T^{9} + 63113778360 T^{10} + 466201213128 T^{11} + 3354742201726 T^{12} + 466201213128 p T^{13} + 63113778360 p^{2} T^{14} + 7718223732 p^{3} T^{15} + 920725379 p^{4} T^{16} + 97798751 p^{5} T^{17} + 10162416 p^{6} T^{18} + 909849 p^{7} T^{19} + 80509 p^{8} T^{20} + 5689 p^{9} T^{21} + 408 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 11 T + 291 T^{2} + 1812 T^{3} + 36937 T^{4} + 160828 T^{5} + 3548923 T^{6} + 12015171 T^{7} + 283246987 T^{8} + 776883390 T^{9} + 19100105770 T^{10} + 45807809032 T^{11} + 1099378225238 T^{12} + 45807809032 p T^{13} + 19100105770 p^{2} T^{14} + 776883390 p^{3} T^{15} + 283246987 p^{4} T^{16} + 12015171 p^{5} T^{17} + 3548923 p^{6} T^{18} + 160828 p^{7} T^{19} + 36937 p^{8} T^{20} + 1812 p^{9} T^{21} + 291 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 8 T + 331 T^{2} + 1786 T^{3} + 48491 T^{4} + 185845 T^{5} + 4709879 T^{6} + 14043911 T^{7} + 368851955 T^{8} + 925316551 T^{9} + 25041043942 T^{10} + 55156822051 T^{11} + 1534727794114 T^{12} + 55156822051 p T^{13} + 25041043942 p^{2} T^{14} + 925316551 p^{3} T^{15} + 368851955 p^{4} T^{16} + 14043911 p^{5} T^{17} + 4709879 p^{6} T^{18} + 185845 p^{7} T^{19} + 48491 p^{8} T^{20} + 1786 p^{9} T^{21} + 331 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 7 T + 593 T^{2} - 3927 T^{3} + 167416 T^{4} - 1031253 T^{5} + 29770556 T^{6} - 168091517 T^{7} + 3713280823 T^{8} - 18955731885 T^{9} + 342199261143 T^{10} - 1555760480836 T^{11} + 23891394948090 T^{12} - 1555760480836 p T^{13} + 342199261143 p^{2} T^{14} - 18955731885 p^{3} T^{15} + 3713280823 p^{4} T^{16} - 168091517 p^{5} T^{17} + 29770556 p^{6} T^{18} - 1031253 p^{7} T^{19} + 167416 p^{8} T^{20} - 3927 p^{9} T^{21} + 593 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 33 T + 955 T^{2} - 18677 T^{3} + 336058 T^{4} - 4968035 T^{5} + 68962144 T^{6} - 837311823 T^{7} + 9636768813 T^{8} - 99692938005 T^{9} + 982655762085 T^{10} - 8814422671130 T^{11} + 75497668846344 T^{12} - 8814422671130 p T^{13} + 982655762085 p^{2} T^{14} - 99692938005 p^{3} T^{15} + 9636768813 p^{4} T^{16} - 837311823 p^{5} T^{17} + 68962144 p^{6} T^{18} - 4968035 p^{7} T^{19} + 336058 p^{8} T^{20} - 18677 p^{9} T^{21} + 955 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 375 T^{2} - 1838 T^{3} + 64529 T^{4} - 695645 T^{5} + 8034081 T^{6} - 118229207 T^{7} + 992593671 T^{8} - 12222243067 T^{9} + 113412835960 T^{10} - 948185851635 T^{11} + 9667401483662 T^{12} - 948185851635 p T^{13} + 113412835960 p^{2} T^{14} - 12222243067 p^{3} T^{15} + 992593671 p^{4} T^{16} - 118229207 p^{5} T^{17} + 8034081 p^{6} T^{18} - 695645 p^{7} T^{19} + 64529 p^{8} T^{20} - 1838 p^{9} T^{21} + 375 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 34 T + 865 T^{2} - 14942 T^{3} + 223224 T^{4} - 2733371 T^{5} + 31197160 T^{6} - 314885302 T^{7} + 3089512385 T^{8} - 27551051130 T^{9} + 245023572825 T^{10} - 2037604887038 T^{11} + 17723144257408 T^{12} - 2037604887038 p T^{13} + 245023572825 p^{2} T^{14} - 27551051130 p^{3} T^{15} + 3089512385 p^{4} T^{16} - 314885302 p^{5} T^{17} + 31197160 p^{6} T^{18} - 2733371 p^{7} T^{19} + 223224 p^{8} T^{20} - 14942 p^{9} T^{21} + 865 p^{10} T^{22} - 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 450 T^{2} - 262 T^{3} + 98960 T^{4} - 175831 T^{5} + 13837924 T^{6} - 50591626 T^{7} + 1390841810 T^{8} - 8678664700 T^{9} + 113179166387 T^{10} - 989933788516 T^{11} + 8746266724000 T^{12} - 989933788516 p T^{13} + 113179166387 p^{2} T^{14} - 8678664700 p^{3} T^{15} + 1390841810 p^{4} T^{16} - 50591626 p^{5} T^{17} + 13837924 p^{6} T^{18} - 175831 p^{7} T^{19} + 98960 p^{8} T^{20} - 262 p^{9} T^{21} + 450 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( 1 - 24 T + 816 T^{2} - 13932 T^{3} + 282810 T^{4} - 3839079 T^{5} + 59254562 T^{6} - 677228411 T^{7} + 8724403259 T^{8} - 87163709779 T^{9} + 985559289534 T^{10} - 8833378136899 T^{11} + 90086503983156 T^{12} - 8833378136899 p T^{13} + 985559289534 p^{2} T^{14} - 87163709779 p^{3} T^{15} + 8724403259 p^{4} T^{16} - 677228411 p^{5} T^{17} + 59254562 p^{6} T^{18} - 3839079 p^{7} T^{19} + 282810 p^{8} T^{20} - 13932 p^{9} T^{21} + 816 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 10 T + 677 T^{2} - 4973 T^{3} + 199963 T^{4} - 970018 T^{5} + 34539545 T^{6} - 77856661 T^{7} + 4029402125 T^{8} + 2230293252 T^{9} + 366800976277 T^{10} + 1086109546196 T^{11} + 31629998271396 T^{12} + 1086109546196 p T^{13} + 366800976277 p^{2} T^{14} + 2230293252 p^{3} T^{15} + 4029402125 p^{4} T^{16} - 77856661 p^{5} T^{17} + 34539545 p^{6} T^{18} - 970018 p^{7} T^{19} + 199963 p^{8} T^{20} - 4973 p^{9} T^{21} + 677 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 16 T + 472 T^{2} - 6274 T^{3} + 115118 T^{4} - 1336097 T^{5} + 18748754 T^{6} - 205495586 T^{7} + 2406005142 T^{8} - 25697485600 T^{9} + 265660085665 T^{10} - 2799949050210 T^{11} + 26707281618992 T^{12} - 2799949050210 p T^{13} + 265660085665 p^{2} T^{14} - 25697485600 p^{3} T^{15} + 2406005142 p^{4} T^{16} - 205495586 p^{5} T^{17} + 18748754 p^{6} T^{18} - 1336097 p^{7} T^{19} + 115118 p^{8} T^{20} - 6274 p^{9} T^{21} + 472 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.48117526328449816225364281940, −2.36471823807202633142799799325, −2.24775538920208487276139467786, −2.20725261169091190172130473361, −2.11821158834300256462578180519, −1.94582216679540801204924583502, −1.80719087120278183721903193480, −1.73275673018029055470973764045, −1.67428317958180831991404313153, −1.57097174490953520786496516893, −1.51382884113010950560121341230, −1.41926412051154135687532193653, −1.41017676843546046134138603112, −1.29474538267094646636577280493, −1.07500710429446259062694613043, −1.02553729585470700711720095992, −0.788863613216677652425481328540, −0.71569389301003497633067728041, −0.71094223461314404699887328990, −0.68342288355487378342540230111, −0.60564849977245159932583855442, −0.57313371352443106563060324869, −0.50656070708796647069115268541, −0.26078318642860361067666544597, −0.15891246357286209866964415513, 0.15891246357286209866964415513, 0.26078318642860361067666544597, 0.50656070708796647069115268541, 0.57313371352443106563060324869, 0.60564849977245159932583855442, 0.68342288355487378342540230111, 0.71094223461314404699887328990, 0.71569389301003497633067728041, 0.788863613216677652425481328540, 1.02553729585470700711720095992, 1.07500710429446259062694613043, 1.29474538267094646636577280493, 1.41017676843546046134138603112, 1.41926412051154135687532193653, 1.51382884113010950560121341230, 1.57097174490953520786496516893, 1.67428317958180831991404313153, 1.73275673018029055470973764045, 1.80719087120278183721903193480, 1.94582216679540801204924583502, 2.11821158834300256462578180519, 2.20725261169091190172130473361, 2.24775538920208487276139467786, 2.36471823807202633142799799325, 2.48117526328449816225364281940
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.