Dirichlet series
| L(s) = 1 | − 5·2-s + 18·3-s − 5·4-s − 12·5-s − 90·6-s − 62·7-s + 55·8-s + 189·9-s + 60·10-s − 72·11-s − 90·12-s − 192·13-s + 310·14-s − 216·15-s − 14·16-s − 100·17-s − 945·18-s − 266·19-s + 60·20-s − 1.11e3·21-s + 360·22-s + 50·23-s + 990·24-s − 306·25-s + 960·26-s + 1.51e3·27-s + 310·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 3.46·3-s − 5/8·4-s − 1.07·5-s − 6.12·6-s − 3.34·7-s + 2.43·8-s + 7·9-s + 1.89·10-s − 1.97·11-s − 2.16·12-s − 4.09·13-s + 5.91·14-s − 3.71·15-s − 0.218·16-s − 1.42·17-s − 12.3·18-s − 3.21·19-s + 0.670·20-s − 11.5·21-s + 3.48·22-s + 0.453·23-s + 8.42·24-s − 2.44·25-s + 7.24·26-s + 10.7·27-s + 2.09·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 67^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 67^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{6} \cdot 67^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.78210\times 10^{6}\) |
| Root analytic conductor: | \(3.44374\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 3^{6} \cdot 67^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T )^{6} \) |
| 67 | \( ( 1 + p T )^{6} \) | |
| good | 2 | \( 1 + 5 T + 15 p T^{2} + 15 p^{3} T^{3} + 489 T^{4} + 1543 T^{5} + 1181 p^{2} T^{6} + 1543 p^{3} T^{7} + 489 p^{6} T^{8} + 15 p^{12} T^{9} + 15 p^{13} T^{10} + 5 p^{15} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 + 12 T + 18 p^{2} T^{2} + 3702 T^{3} + 84996 T^{4} + 551406 T^{5} + 11272786 T^{6} + 551406 p^{3} T^{7} + 84996 p^{6} T^{8} + 3702 p^{9} T^{9} + 18 p^{14} T^{10} + 12 p^{15} T^{11} + p^{18} T^{12} \) | |
| 7 | \( 1 + 62 T + 3102 T^{2} + 105906 T^{3} + 3095740 T^{4} + 72288890 T^{5} + 1474626964 T^{6} + 72288890 p^{3} T^{7} + 3095740 p^{6} T^{8} + 105906 p^{9} T^{9} + 3102 p^{12} T^{10} + 62 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 72 T + 7730 T^{2} + 383004 T^{3} + 2199229 p T^{4} + 914116548 T^{5} + 42016412684 T^{6} + 914116548 p^{3} T^{7} + 2199229 p^{7} T^{8} + 383004 p^{9} T^{9} + 7730 p^{12} T^{10} + 72 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 192 T + 24192 T^{2} + 2178520 T^{3} + 159997191 T^{4} + 9607642896 T^{5} + 491628309456 T^{6} + 9607642896 p^{3} T^{7} + 159997191 p^{6} T^{8} + 2178520 p^{9} T^{9} + 24192 p^{12} T^{10} + 192 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 100 T + 17406 T^{2} + 1324680 T^{3} + 167225871 T^{4} + 10618927604 T^{5} + 1002763591076 T^{6} + 10618927604 p^{3} T^{7} + 167225871 p^{6} T^{8} + 1324680 p^{9} T^{9} + 17406 p^{12} T^{10} + 100 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 14 p T + 42266 T^{2} + 4236138 T^{3} + 296092135 T^{4} + 705418408 p T^{5} + 719366719084 T^{6} + 705418408 p^{4} T^{7} + 296092135 p^{6} T^{8} + 4236138 p^{9} T^{9} + 42266 p^{12} T^{10} + 14 p^{16} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 50 T + 59580 T^{2} - 2453988 T^{3} + 1563496296 T^{4} - 52610439040 T^{5} + 24008793796256 T^{6} - 52610439040 p^{3} T^{7} + 1563496296 p^{6} T^{8} - 2453988 p^{9} T^{9} + 59580 p^{12} T^{10} - 50 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 242 T + 110184 T^{2} + 21008874 T^{3} + 5348102751 T^{4} + 825044232844 T^{5} + 158910112031072 T^{6} + 825044232844 p^{3} T^{7} + 5348102751 p^{6} T^{8} + 21008874 p^{9} T^{9} + 110184 p^{12} T^{10} + 242 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 + 438 T + 167178 T^{2} + 47886674 T^{3} + 11737875204 T^{4} + 2476948325742 T^{5} + 452905537959060 T^{6} + 2476948325742 p^{3} T^{7} + 11737875204 p^{6} T^{8} + 47886674 p^{9} T^{9} + 167178 p^{12} T^{10} + 438 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 596 T + 371988 T^{2} + 141686280 T^{3} + 51370265512 T^{4} + 13921624565720 T^{5} + 3556511420813206 T^{6} + 13921624565720 p^{3} T^{7} + 51370265512 p^{6} T^{8} + 141686280 p^{9} T^{9} + 371988 p^{12} T^{10} + 596 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 54 T + 161642 T^{2} - 8481930 T^{3} + 10616368448 T^{4} - 1249043400036 T^{5} + 591220917557702 T^{6} - 1249043400036 p^{3} T^{7} + 10616368448 p^{6} T^{8} - 8481930 p^{9} T^{9} + 161642 p^{12} T^{10} - 54 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 360 T + 165516 T^{2} + 47995920 T^{3} + 20642421372 T^{4} + 5792529905856 T^{5} + 2041194096923078 T^{6} + 5792529905856 p^{3} T^{7} + 20642421372 p^{6} T^{8} + 47995920 p^{9} T^{9} + 165516 p^{12} T^{10} + 360 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 720 T + 397004 T^{2} + 125695416 T^{3} + 26888555879 T^{4} + 2268069480336 T^{5} - 180545365290088 T^{6} + 2268069480336 p^{3} T^{7} + 26888555879 p^{6} T^{8} + 125695416 p^{9} T^{9} + 397004 p^{12} T^{10} + 720 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 694 T + 664990 T^{2} - 406886710 T^{3} + 231435834356 T^{4} - 103855499414956 T^{5} + 45785711849399818 T^{6} - 103855499414956 p^{3} T^{7} + 231435834356 p^{6} T^{8} - 406886710 p^{9} T^{9} + 664990 p^{12} T^{10} - 694 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 378 T + 704502 T^{2} + 120948282 T^{3} + 212924962980 T^{4} + 17102539469628 T^{5} + 48061764657521098 T^{6} + 17102539469628 p^{3} T^{7} + 212924962980 p^{6} T^{8} + 120948282 p^{9} T^{9} + 704502 p^{12} T^{10} + 378 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 1396 T + 1977224 T^{2} + 1627616532 T^{3} + 1297419985615 T^{4} + 740967503983568 T^{5} + 408223248984371776 T^{6} + 740967503983568 p^{3} T^{7} + 1297419985615 p^{6} T^{8} + 1627616532 p^{9} T^{9} + 1977224 p^{12} T^{10} + 1396 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 964 T + 1676964 T^{2} + 1220182788 T^{3} + 1146291805335 T^{4} + 691552394370056 T^{5} + 483190046029632872 T^{6} + 691552394370056 p^{3} T^{7} + 1146291805335 p^{6} T^{8} + 1220182788 p^{9} T^{9} + 1676964 p^{12} T^{10} + 964 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 192 T + 835608 T^{2} + 301915460 T^{3} + 533817926664 T^{4} + 150219534958800 T^{5} + 258543945814639926 T^{6} + 150219534958800 p^{3} T^{7} + 533817926664 p^{6} T^{8} + 301915460 p^{9} T^{9} + 835608 p^{12} T^{10} + 192 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 802 T + 1938044 T^{2} + 1472087886 T^{3} + 2006062790071 T^{4} + 1237567282690604 T^{5} + 1248751754152488280 T^{6} + 1237567282690604 p^{3} T^{7} + 2006062790071 p^{6} T^{8} + 1472087886 p^{9} T^{9} + 1938044 p^{12} T^{10} + 802 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 2126 T + 3463794 T^{2} - 4147751538 T^{3} + 4417404883020 T^{4} - 3937769393691868 T^{5} + 3191365359439157882 T^{6} - 3937769393691868 p^{3} T^{7} + 4417404883020 p^{6} T^{8} - 4147751538 p^{9} T^{9} + 3463794 p^{12} T^{10} - 2126 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 - 432 T + 1092362 T^{2} - 1355191524 T^{3} + 1512520351031 T^{4} - 1154979184953420 T^{5} + 1362263626157437724 T^{6} - 1154979184953420 p^{3} T^{7} + 1512520351031 p^{6} T^{8} - 1355191524 p^{9} T^{9} + 1092362 p^{12} T^{10} - 432 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 1290 T + 3920922 T^{2} - 4779758798 T^{3} + 7817886084711 T^{4} - 77348544875976 p T^{5} + 9271207523010554172 T^{6} - 77348544875976 p^{4} T^{7} + 7817886084711 p^{6} T^{8} - 4779758798 p^{9} T^{9} + 3920922 p^{12} T^{10} - 1290 p^{15} T^{11} + p^{18} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29221667386695531969572842517, −6.94104998703106854346167888621, −6.75281102455023465571879884762, −6.61874301275825409927809978204, −6.21897198183352610820356630677, −6.13518667566448620679016942111, −6.08558343637590145743237593582, −5.54996425670707457018266274619, −5.03549845706702368157840730440, −4.91576170977704523423493943087, −4.89210195200042276036866401283, −4.68104910801038599570570326404, −4.54311359391929379066814004795, −4.10903919175663666728746486171, −3.70166069189759412580127361959, −3.65892642364710805721816545671, −3.57560639691501326042824849429, −3.53072541613820279902345194393, −2.97765857280922206642966364540, −2.74196089265084317987924167886, −2.67223665061093574358940742205, −2.33958587247743485164368162051, −2.02918787600409761608063260615, −1.86769088857789682107879958772, −1.74349042328075579995667823048, 0, 0, 0, 0, 0, 0, 1.74349042328075579995667823048, 1.86769088857789682107879958772, 2.02918787600409761608063260615, 2.33958587247743485164368162051, 2.67223665061093574358940742205, 2.74196089265084317987924167886, 2.97765857280922206642966364540, 3.53072541613820279902345194393, 3.57560639691501326042824849429, 3.65892642364710805721816545671, 3.70166069189759412580127361959, 4.10903919175663666728746486171, 4.54311359391929379066814004795, 4.68104910801038599570570326404, 4.89210195200042276036866401283, 4.91576170977704523423493943087, 5.03549845706702368157840730440, 5.54996425670707457018266274619, 6.08558343637590145743237593582, 6.13518667566448620679016942111, 6.21897198183352610820356630677, 6.61874301275825409927809978204, 6.75281102455023465571879884762, 6.94104998703106854346167888621, 7.29221667386695531969572842517
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.