Dirichlet series
| L(s) = 1 | + 2·2-s − 15·4-s + 30·5-s − 16·8-s + 60·10-s + 16·11-s + 168·13-s + 191·16-s + 4·17-s + 308·19-s − 450·20-s + 32·22-s + 336·23-s + 525·25-s + 336·26-s − 176·29-s + 392·31-s + 158·32-s + 8·34-s − 140·37-s + 616·38-s − 480·40-s − 656·41-s − 388·43-s − 240·44-s + 672·46-s − 628·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.87·4-s + 2.68·5-s − 0.707·8-s + 1.89·10-s + 0.438·11-s + 3.58·13-s + 2.98·16-s + 0.0570·17-s + 3.71·19-s − 5.03·20-s + 0.310·22-s + 3.04·23-s + 21/5·25-s + 2.53·26-s − 1.12·29-s + 2.27·31-s + 0.872·32-s + 0.0403·34-s − 0.622·37-s + 2.62·38-s − 1.89·40-s − 2.49·41-s − 1.37·43-s − 0.822·44-s + 2.15·46-s − 1.94·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{12} \cdot 5^{6} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.84895\times 10^{12}\) |
| Root analytic conductor: | \(11.4061\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | induced by $\chi_{2205} (1, \cdot )$ |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 3^{12} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(178.8135298\) |
| \(L(\frac12)\) | \(\approx\) | \(178.8135298\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - p T )^{6} \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 - p T + 19 T^{2} - 13 p^{2} T^{3} + 83 p T^{4} - 73 p^{3} T^{5} + 305 p^{2} T^{6} - 73 p^{6} T^{7} + 83 p^{7} T^{8} - 13 p^{11} T^{9} + 19 p^{12} T^{10} - p^{16} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 16 T + 812 T^{2} + 15300 T^{3} + 2567184 T^{4} + 56858680 T^{5} - 552711974 T^{6} + 56858680 p^{3} T^{7} + 2567184 p^{6} T^{8} + 15300 p^{9} T^{9} + 812 p^{12} T^{10} - 16 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 168 T + 18956 T^{2} - 116156 p T^{3} + 104466904 T^{4} - 6040563440 T^{5} + 309756529486 T^{6} - 6040563440 p^{3} T^{7} + 104466904 p^{6} T^{8} - 116156 p^{10} T^{9} + 18956 p^{12} T^{10} - 168 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - 4 T + 14086 T^{2} + 160336 T^{3} + 115923452 T^{4} + 1228657188 T^{5} + 692428906716 T^{6} + 1228657188 p^{3} T^{7} + 115923452 p^{6} T^{8} + 160336 p^{9} T^{9} + 14086 p^{12} T^{10} - 4 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 308 T + 73404 T^{2} - 12077372 T^{3} + 1625488783 T^{4} - 176451707096 T^{5} + 16038736148248 T^{6} - 176451707096 p^{3} T^{7} + 1625488783 p^{6} T^{8} - 12077372 p^{9} T^{9} + 73404 p^{12} T^{10} - 308 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 336 T + 98630 T^{2} - 19185056 T^{3} + 3349928211 T^{4} - 19757853136 p T^{5} + 105705477964 p^{2} T^{6} - 19757853136 p^{4} T^{7} + 3349928211 p^{6} T^{8} - 19185056 p^{9} T^{9} + 98630 p^{12} T^{10} - 336 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 176 T + 92656 T^{2} + 12546380 T^{3} + 4288608312 T^{4} + 478053343680 T^{5} + 127431913139622 T^{6} + 478053343680 p^{3} T^{7} + 4288608312 p^{6} T^{8} + 12546380 p^{9} T^{9} + 92656 p^{12} T^{10} + 176 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 392 T + 153030 T^{2} - 32287912 T^{3} + 7677507203 T^{4} - 1249077738464 T^{5} + 255230995151820 T^{6} - 1249077738464 p^{3} T^{7} + 7677507203 p^{6} T^{8} - 32287912 p^{9} T^{9} + 153030 p^{12} T^{10} - 392 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 140 T + 213942 T^{2} + 12843948 T^{3} + 18729025499 T^{4} + 132721017192 T^{5} + 1059724116139756 T^{6} + 132721017192 p^{3} T^{7} + 18729025499 p^{6} T^{8} + 12843948 p^{9} T^{9} + 213942 p^{12} T^{10} + 140 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 16 p T + 421976 T^{2} + 166159344 T^{3} + 65879968003 T^{4} + 19715169399936 T^{5} + 5871622216065200 T^{6} + 19715169399936 p^{3} T^{7} + 65879968003 p^{6} T^{8} + 166159344 p^{9} T^{9} + 421976 p^{12} T^{10} + 16 p^{16} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 388 T + 421158 T^{2} + 116550300 T^{3} + 73177187783 T^{4} + 15517478343240 T^{5} + 7317237698378164 T^{6} + 15517478343240 p^{3} T^{7} + 73177187783 p^{6} T^{8} + 116550300 p^{9} T^{9} + 421158 p^{12} T^{10} + 388 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 628 T + 589452 T^{2} + 246437960 T^{3} + 139029029008 T^{4} + 44711775028180 T^{5} + 18650987083950686 T^{6} + 44711775028180 p^{3} T^{7} + 139029029008 p^{6} T^{8} + 246437960 p^{9} T^{9} + 589452 p^{12} T^{10} + 628 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 676 T + 756806 T^{2} - 367769844 T^{3} + 249160429851 T^{4} - 95287278237352 T^{5} + 47830207170007020 T^{6} - 95287278237352 p^{3} T^{7} + 249160429851 p^{6} T^{8} - 367769844 p^{9} T^{9} + 756806 p^{12} T^{10} - 676 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 996 T + 1036536 T^{2} + 686090796 T^{3} + 452936806907 T^{4} + 228269279812616 T^{5} + 117358818568794736 T^{6} + 228269279812616 p^{3} T^{7} + 452936806907 p^{6} T^{8} + 686090796 p^{9} T^{9} + 1036536 p^{12} T^{10} + 996 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 740 T + 970482 T^{2} - 661891780 T^{3} + 495713522631 T^{4} - 257223694985640 T^{5} + 147582202110218748 T^{6} - 257223694985640 p^{3} T^{7} + 495713522631 p^{6} T^{8} - 661891780 p^{9} T^{9} + 970482 p^{12} T^{10} - 740 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 1768 T + 2799902 T^{2} - 2855569976 T^{3} + 2569585021227 T^{4} - 1777311287531536 T^{5} + 1093536943175084732 T^{6} - 1777311287531536 p^{3} T^{7} + 2569585021227 p^{6} T^{8} - 2855569976 p^{9} T^{9} + 2799902 p^{12} T^{10} - 1768 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 224 T + 735802 T^{2} + 33746048 T^{3} + 362779099663 T^{4} - 5782399589792 T^{5} + 175695286389376652 T^{6} - 5782399589792 p^{3} T^{7} + 362779099663 p^{6} T^{8} + 33746048 p^{9} T^{9} + 735802 p^{12} T^{10} - 224 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 2640 T + 4127428 T^{2} - 4281599440 T^{3} + 3477156414455 T^{4} - 2335723050987360 T^{5} + 1487916107494354120 T^{6} - 2335723050987360 p^{3} T^{7} + 3477156414455 p^{6} T^{8} - 4281599440 p^{9} T^{9} + 4127428 p^{12} T^{10} - 2640 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 1636 T + 3307008 T^{2} - 41764072 p T^{3} + 3796696094800 T^{4} - 2753755692366396 T^{5} + 2363360511013187334 T^{6} - 2753755692366396 p^{3} T^{7} + 3796696094800 p^{6} T^{8} - 41764072 p^{10} T^{9} + 3307008 p^{12} T^{10} - 1636 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 + 140 T + 1262996 T^{2} - 466318380 T^{3} + 677854086191 T^{4} - 649780965410440 T^{5} + 346141233175430792 T^{6} - 649780965410440 p^{3} T^{7} + 677854086191 p^{6} T^{8} - 466318380 p^{9} T^{9} + 1262996 p^{12} T^{10} + 140 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 - 1904 T + 3824692 T^{2} - 4929406096 T^{3} + 6160110405943 T^{4} - 5994022973389568 T^{5} + 5629689575835156904 T^{6} - 5994022973389568 p^{3} T^{7} + 6160110405943 p^{6} T^{8} - 4929406096 p^{9} T^{9} + 3824692 p^{12} T^{10} - 1904 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 516 T + 3952326 T^{2} - 925377376 T^{3} + 6580438313612 T^{4} - 355370807466588 T^{5} + 6947174177634861996 T^{6} - 355370807466588 p^{3} T^{7} + 6580438313612 p^{6} T^{8} - 925377376 p^{9} T^{9} + 3952326 p^{12} T^{10} - 516 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−4.53495372954292475609337195101, −3.94301767491994167299432621593, −3.82334162062850331270272523718, −3.77412055282495582050677978420, −3.71231190918797814094073408801, −3.68061146576806708701774357429, −3.62148541064919988017290443933, −3.23306330096569356934323660889, −3.21888289800798435304563618631, −2.95120602018056173510434217557, −2.92284695828335124406143698803, −2.74796657112291585281408525453, −2.44811561400492220998719715751, −2.16717496199725693700868241537, −1.97525490437392965992526973694, −1.82684102205466336176243864705, −1.61535225736761742405203726089, −1.40934051512823791689215379975, −1.32733214524818782582278840857, −1.22952304195467700663502412052, −0.848489568283673992685265416503, −0.846889252929135547993845561864, −0.77289365785794253476888403200, −0.51523972753236049132633325599, −0.46085740632888487602078264595, 0.46085740632888487602078264595, 0.51523972753236049132633325599, 0.77289365785794253476888403200, 0.846889252929135547993845561864, 0.848489568283673992685265416503, 1.22952304195467700663502412052, 1.32733214524818782582278840857, 1.40934051512823791689215379975, 1.61535225736761742405203726089, 1.82684102205466336176243864705, 1.97525490437392965992526973694, 2.16717496199725693700868241537, 2.44811561400492220998719715751, 2.74796657112291585281408525453, 2.92284695828335124406143698803, 2.95120602018056173510434217557, 3.21888289800798435304563618631, 3.23306330096569356934323660889, 3.62148541064919988017290443933, 3.68061146576806708701774357429, 3.71231190918797814094073408801, 3.77412055282495582050677978420, 3.82334162062850331270272523718, 3.94301767491994167299432621593, 4.53495372954292475609337195101
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.