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: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 3^{12} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
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 | \( 1 \) |
| 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
−4.87312129884779533239193993492, −4.49312024844378317195099888867, −4.39366655066510195516184319763, −4.30462885022467551900177280186, −4.17027690025449907385227268342, −4.15682767688088758323016120358, −4.02971544017689490587494306976, −3.76714144885702383325229105485, −3.59171711170322547995873872642, −3.55914854678792437858934322195, −3.50525396743630948966476999391, −3.19451876207692627987878524009, −3.10807392973969823565264596720, −2.64430203788829778471718006019, −2.57163333862389069992405359272, −2.44600663353701162806966015844, −2.36068377932812729027819785294, −2.26086568436114719232449294986, −1.95110997524623837962127854613, −1.89957176120048948966404910956, −1.30509070600569735798743043172, −1.03782097414346143926121900093, −1.00441889402860758971246066048, −0.998178678086866782112530470345, −0.854416625981987310988654899825, 0, 0, 0, 0, 0, 0, 0.854416625981987310988654899825, 0.998178678086866782112530470345, 1.00441889402860758971246066048, 1.03782097414346143926121900093, 1.30509070600569735798743043172, 1.89957176120048948966404910956, 1.95110997524623837962127854613, 2.26086568436114719232449294986, 2.36068377932812729027819785294, 2.44600663353701162806966015844, 2.57163333862389069992405359272, 2.64430203788829778471718006019, 3.10807392973969823565264596720, 3.19451876207692627987878524009, 3.50525396743630948966476999391, 3.55914854678792437858934322195, 3.59171711170322547995873872642, 3.76714144885702383325229105485, 4.02971544017689490587494306976, 4.15682767688088758323016120358, 4.17027690025449907385227268342, 4.30462885022467551900177280186, 4.39366655066510195516184319763, 4.49312024844378317195099888867, 4.87312129884779533239193993492
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.