Dirichlet series
| L(s) = 1 | + 3·2-s + 3·3-s + 3·4-s − 3·5-s + 9·6-s − 3·7-s + 8-s + 12·9-s − 9·10-s − 11-s + 9·12-s + 2·13-s − 9·14-s − 9·15-s + 7·17-s + 36·18-s + 6·19-s − 9·20-s − 9·21-s − 3·22-s + 8·23-s + 3·24-s + 3·25-s + 6·26-s + 28·27-s − 9·28-s + 20·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3/2·4-s − 1.34·5-s + 3.67·6-s − 1.13·7-s + 0.353·8-s + 4·9-s − 2.84·10-s − 0.301·11-s + 2.59·12-s + 0.554·13-s − 2.40·14-s − 2.32·15-s + 1.69·17-s + 8.48·18-s + 1.37·19-s − 2.01·20-s − 1.96·21-s − 0.639·22-s + 1.66·23-s + 0.612·24-s + 3/5·25-s + 1.17·26-s + 5.38·27-s − 1.70·28-s + 3.71·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 7^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 7^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{12} \cdot 7^{12} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.91886\times 10^{9}\) |
| Root analytic conductor: | \(2.47961\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{12} \cdot 7^{12} \cdot 11^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(127.2547985\) |
| \(L(\frac12)\) | \(\approx\) | \(127.2547985\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \) | |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \) | |
| 11 | \( 1 + T - 18 T^{2} + 12 T^{3} + 193 T^{4} - 18 p T^{5} - 2187 T^{6} - 18 p^{2} T^{7} + 193 p^{2} T^{8} + 12 p^{3} T^{9} - 18 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 - p T - p T^{2} + 17 T^{3} - 7 T^{4} - 20 T^{5} + 10 T^{6} - 44 T^{7} + 62 p T^{8} - 140 T^{9} - 194 T^{10} + 548 T^{11} - 1034 T^{12} + 548 p T^{13} - 194 p^{2} T^{14} - 140 p^{3} T^{15} + 62 p^{5} T^{16} - 44 p^{5} T^{17} + 10 p^{6} T^{18} - 20 p^{7} T^{19} - 7 p^{8} T^{20} + 17 p^{9} T^{21} - p^{11} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 13 | \( 1 - 2 T - 3 T^{2} + 6 T^{3} + 326 T^{4} - 296 T^{5} - 3332 T^{6} + 1944 T^{7} + 47777 T^{8} + 10432 T^{9} - 1174294 T^{10} + 682920 T^{11} + 4749735 T^{12} + 682920 p T^{13} - 1174294 p^{2} T^{14} + 10432 p^{3} T^{15} + 47777 p^{4} T^{16} + 1944 p^{5} T^{17} - 3332 p^{6} T^{18} - 296 p^{7} T^{19} + 326 p^{8} T^{20} + 6 p^{9} T^{21} - 3 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 7 T - 15 T^{2} + 249 T^{3} - 603 T^{4} - 960 T^{5} + 8824 T^{6} - 28552 T^{7} + 119140 T^{8} - 216576 T^{9} - 1976934 T^{10} + 5973666 T^{11} + 8202062 T^{12} + 5973666 p T^{13} - 1976934 p^{2} T^{14} - 216576 p^{3} T^{15} + 119140 p^{4} T^{16} - 28552 p^{5} T^{17} + 8824 p^{6} T^{18} - 960 p^{7} T^{19} - 603 p^{8} T^{20} + 249 p^{9} T^{21} - 15 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 6 T - 52 T^{2} + 560 T^{3} - 21 T^{4} - 938 p T^{5} + 63996 T^{6} + 157836 T^{7} - 1681264 T^{8} + 3464626 T^{9} + 8102308 T^{10} - 57833206 T^{11} + 204749440 T^{12} - 57833206 p T^{13} + 8102308 p^{2} T^{14} + 3464626 p^{3} T^{15} - 1681264 p^{4} T^{16} + 157836 p^{5} T^{17} + 63996 p^{6} T^{18} - 938 p^{8} T^{19} - 21 p^{8} T^{20} + 560 p^{9} T^{21} - 52 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 4 T + 2 p T^{2} - 300 T^{3} + 1759 T^{4} - 7624 T^{5} + 53956 T^{6} - 7624 p T^{7} + 1759 p^{2} T^{8} - 300 p^{3} T^{9} + 2 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 - 20 T + 83 T^{2} + 1066 T^{3} - 12544 T^{4} + 19686 T^{5} + 489146 T^{6} - 3569234 T^{7} - 2097381 T^{8} + 148678402 T^{9} - 21086848 p T^{10} - 2136186562 T^{11} + 28311694255 T^{12} - 2136186562 p T^{13} - 21086848 p^{3} T^{14} + 148678402 p^{3} T^{15} - 2097381 p^{4} T^{16} - 3569234 p^{5} T^{17} + 489146 p^{6} T^{18} + 19686 p^{7} T^{19} - 12544 p^{8} T^{20} + 1066 p^{9} T^{21} + 83 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 6 T - 73 T^{2} + 160 T^{3} + 3950 T^{4} - 1190 T^{5} - 50348 T^{6} - 292438 T^{7} - 1197207 T^{8} + 510750 T^{9} + 145550070 T^{10} - 76602346 T^{11} - 4142768345 T^{12} - 76602346 p T^{13} + 145550070 p^{2} T^{14} + 510750 p^{3} T^{15} - 1197207 p^{4} T^{16} - 292438 p^{5} T^{17} - 50348 p^{6} T^{18} - 1190 p^{7} T^{19} + 3950 p^{8} T^{20} + 160 p^{9} T^{21} - 73 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 22 T + 191 T^{2} - 968 T^{3} + 6556 T^{4} - 39914 T^{5} - 66342 T^{6} + 1934170 T^{7} - 6442321 T^{8} + 38354846 T^{9} - 516072000 T^{10} + 1284289862 T^{11} + 6603726451 T^{12} + 1284289862 p T^{13} - 516072000 p^{2} T^{14} + 38354846 p^{3} T^{15} - 6442321 p^{4} T^{16} + 1934170 p^{5} T^{17} - 66342 p^{6} T^{18} - 39914 p^{7} T^{19} + 6556 p^{8} T^{20} - 968 p^{9} T^{21} + 191 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 2 T - 51 T^{2} - 188 T^{3} + 1346 T^{4} - 28382 T^{5} + 76982 T^{6} + 1644890 T^{7} + 5383467 T^{8} - 40315358 T^{9} + 357217854 T^{10} - 1484471310 T^{11} - 23702864577 T^{12} - 1484471310 p T^{13} + 357217854 p^{2} T^{14} - 40315358 p^{3} T^{15} + 5383467 p^{4} T^{16} + 1644890 p^{5} T^{17} + 76982 p^{6} T^{18} - 28382 p^{7} T^{19} + 1346 p^{8} T^{20} - 188 p^{9} T^{21} - 51 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 30 T + 513 T^{2} + 6248 T^{3} + 60360 T^{4} + 486492 T^{5} + 3407896 T^{6} + 486492 p T^{7} + 60360 p^{2} T^{8} + 6248 p^{3} T^{9} + 513 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 4 T - 75 T^{2} + 62 T^{3} + 7040 T^{4} - 4278 T^{5} - 374854 T^{6} + 936066 T^{7} + 15629457 T^{8} - 62620302 T^{9} - 730390838 T^{10} + 137238050 T^{11} + 22540924105 T^{12} + 137238050 p T^{13} - 730390838 p^{2} T^{14} - 62620302 p^{3} T^{15} + 15629457 p^{4} T^{16} + 936066 p^{5} T^{17} - 374854 p^{6} T^{18} - 4278 p^{7} T^{19} + 7040 p^{8} T^{20} + 62 p^{9} T^{21} - 75 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 18 T - 13 T^{2} + 2250 T^{3} - 7240 T^{4} - 172436 T^{5} + 994190 T^{6} + 10203884 T^{7} - 90717521 T^{8} - 425367964 T^{9} + 6353962848 T^{10} + 9251744192 T^{11} - 376550087169 T^{12} + 9251744192 p T^{13} + 6353962848 p^{2} T^{14} - 425367964 p^{3} T^{15} - 90717521 p^{4} T^{16} + 10203884 p^{5} T^{17} + 994190 p^{6} T^{18} - 172436 p^{7} T^{19} - 7240 p^{8} T^{20} + 2250 p^{9} T^{21} - 13 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 32 T + 364 T^{2} + 1250 T^{3} - 641 T^{4} + 131636 T^{5} + 1733532 T^{6} - 1465908 T^{7} - 98415680 T^{8} + 316171672 T^{9} + 10289079204 T^{10} + 22571288718 T^{11} - 275683944744 T^{12} + 22571288718 p T^{13} + 10289079204 p^{2} T^{14} + 316171672 p^{3} T^{15} - 98415680 p^{4} T^{16} - 1465908 p^{5} T^{17} + 1733532 p^{6} T^{18} + 131636 p^{7} T^{19} - 641 p^{8} T^{20} + 1250 p^{9} T^{21} + 364 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 8 T - 165 T^{2} + 198 T^{3} + 21084 T^{4} + 95278 T^{5} - 1475824 T^{6} - 14364650 T^{7} + 21948789 T^{8} + 1047689262 T^{9} + 4810375720 T^{10} - 30675501510 T^{11} - 447538517245 T^{12} - 30675501510 p T^{13} + 4810375720 p^{2} T^{14} + 1047689262 p^{3} T^{15} + 21948789 p^{4} T^{16} - 14364650 p^{5} T^{17} - 1475824 p^{6} T^{18} + 95278 p^{7} T^{19} + 21084 p^{8} T^{20} + 198 p^{9} T^{21} - 165 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 - 18 T + 417 T^{2} - 5348 T^{3} + 69572 T^{4} - 671932 T^{5} + 6180304 T^{6} - 671932 p T^{7} + 69572 p^{2} T^{8} - 5348 p^{3} T^{9} + 417 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 + 34 T + 415 T^{2} + 972 T^{3} - 29724 T^{4} - 391842 T^{5} - 2494392 T^{6} - 12270726 T^{7} - 108072957 T^{8} - 1095891670 T^{9} + 762467890 T^{10} + 194216107858 T^{11} + 2532058684637 T^{12} + 194216107858 p T^{13} + 762467890 p^{2} T^{14} - 1095891670 p^{3} T^{15} - 108072957 p^{4} T^{16} - 12270726 p^{5} T^{17} - 2494392 p^{6} T^{18} - 391842 p^{7} T^{19} - 29724 p^{8} T^{20} + 972 p^{9} T^{21} + 415 p^{10} T^{22} + 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 14 T - 115 T^{2} + 3280 T^{3} - 5622 T^{4} - 276622 T^{5} + 1228462 T^{6} + 16347550 T^{7} - 70468773 T^{8} - 1500535118 T^{9} + 11703091646 T^{10} + 67114025778 T^{11} - 1380063119561 T^{12} + 67114025778 p T^{13} + 11703091646 p^{2} T^{14} - 1500535118 p^{3} T^{15} - 70468773 p^{4} T^{16} + 16347550 p^{5} T^{17} + 1228462 p^{6} T^{18} - 276622 p^{7} T^{19} - 5622 p^{8} T^{20} + 3280 p^{9} T^{21} - 115 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 12 T - 35 T^{2} + 112 T^{3} + 11138 T^{4} + 17416 T^{5} + 247050 T^{6} + 1433216 T^{7} - 38765767 T^{8} + 506886364 T^{9} + 6488882120 T^{10} + 8288472424 T^{11} + 46251172077 T^{12} + 8288472424 p T^{13} + 6488882120 p^{2} T^{14} + 506886364 p^{3} T^{15} - 38765767 p^{4} T^{16} + 1433216 p^{5} T^{17} + 247050 p^{6} T^{18} + 17416 p^{7} T^{19} + 11138 p^{8} T^{20} + 112 p^{9} T^{21} - 35 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 30 T + 322 T^{2} - 386 T^{3} - 24805 T^{4} + 311666 T^{5} - 2551554 T^{6} + 19759016 T^{7} - 151650278 T^{8} + 1073133334 T^{9} + 3622329992 T^{10} - 276341228340 T^{11} + 3796552967460 T^{12} - 276341228340 p T^{13} + 3622329992 p^{2} T^{14} + 1073133334 p^{3} T^{15} - 151650278 p^{4} T^{16} + 19759016 p^{5} T^{17} - 2551554 p^{6} T^{18} + 311666 p^{7} T^{19} - 24805 p^{8} T^{20} - 386 p^{9} T^{21} + 322 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 18 T + 499 T^{2} + 7566 T^{3} + 106924 T^{4} + 1309542 T^{5} + 12534832 T^{6} + 1309542 p T^{7} + 106924 p^{2} T^{8} + 7566 p^{3} T^{9} + 499 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 39 T + 718 T^{2} - 8470 T^{3} + 87740 T^{4} - 1006853 T^{5} + 9796705 T^{6} - 45304492 T^{7} - 282230971 T^{8} + 7489239063 T^{9} - 109513216463 T^{10} + 1587053833399 T^{11} - 18491630528964 T^{12} + 1587053833399 p T^{13} - 109513216463 p^{2} T^{14} + 7489239063 p^{3} T^{15} - 282230971 p^{4} T^{16} - 45304492 p^{5} T^{17} + 9796705 p^{6} T^{18} - 1006853 p^{7} T^{19} + 87740 p^{8} T^{20} - 8470 p^{9} T^{21} + 718 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.38987808916570256324063870935, −3.20601959485726719015000418453, −3.15139869836321740933095976900, −3.13968874353076293368353828585, −3.08496948289144095536127658995, −3.07034420254592670669583017108, −3.00894854158764046733356370579, −2.78325421691987277707480550126, −2.60025199134365500048133350244, −2.46777985945349701545628240215, −2.38512738202390087968333327758, −2.28610606518893362117725516724, −2.11294237574936178537463602773, −1.94910235144953301933116409020, −1.90536624329982936118739202942, −1.85154599040212934216473961993, −1.51966632161400432708982381953, −1.44378071889513271861018807048, −1.29480050600874330402203066847, −1.26685381403987682378336485733, −1.09737667956544745149517158187, −0.76652484081228053478780132746, −0.65832266507572345647934388844, −0.64042516928844075467022594169, −0.46245590216285114959954023030, 0.46245590216285114959954023030, 0.64042516928844075467022594169, 0.65832266507572345647934388844, 0.76652484081228053478780132746, 1.09737667956544745149517158187, 1.26685381403987682378336485733, 1.29480050600874330402203066847, 1.44378071889513271861018807048, 1.51966632161400432708982381953, 1.85154599040212934216473961993, 1.90536624329982936118739202942, 1.94910235144953301933116409020, 2.11294237574936178537463602773, 2.28610606518893362117725516724, 2.38512738202390087968333327758, 2.46777985945349701545628240215, 2.60025199134365500048133350244, 2.78325421691987277707480550126, 3.00894854158764046733356370579, 3.07034420254592670669583017108, 3.08496948289144095536127658995, 3.13968874353076293368353828585, 3.15139869836321740933095976900, 3.20601959485726719015000418453, 3.38987808916570256324063870935
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.