Dirichlet series
| L(s) = 1 | + 12·2-s + 78·4-s + 364·8-s − 4·11-s + 1.36e3·16-s + 16·17-s − 48·22-s + 28·25-s + 16·29-s + 4.36e3·32-s + 192·34-s + 24·37-s + 16·41-s − 312·44-s − 6·49-s + 336·50-s + 192·58-s + 1.23e4·64-s − 48·67-s + 1.24e3·68-s + 288·74-s + 192·82-s + 16·83-s − 1.45e3·88-s + 48·97-s − 72·98-s + 2.18e3·100-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 39·4-s + 128.·8-s − 1.20·11-s + 341.·16-s + 3.88·17-s − 10.2·22-s + 28/5·25-s + 2.97·29-s + 772.·32-s + 32.9·34-s + 3.94·37-s + 2.49·41-s − 47.0·44-s − 6/7·49-s + 47.5·50-s + 25.2·58-s + 1.54e3·64-s − 5.86·67-s + 151.·68-s + 33.4·74-s + 21.2·82-s + 1.75·83-s − 155.·88-s + 4.87·97-s − 7.27·98-s + 218.·100-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.37663\times 10^{12}\) |
| Root analytic conductor: | \(3.32675\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 11^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(52391.90663\) |
| \(L(\frac12)\) | \(\approx\) | \(52391.90663\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{12} \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 + T^{2} )^{6} \) | |
| 11 | \( 1 + 4 T + 6 T^{2} + 28 T^{3} - 57 T^{4} - 472 T^{5} - 556 T^{6} - 472 p T^{7} - 57 p^{2} T^{8} + 28 p^{3} T^{9} + 6 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 5 | \( 1 - 28 T^{2} + 406 T^{4} - 828 p T^{6} + 33203 T^{8} - 43592 p T^{10} + 1190956 T^{12} - 43592 p^{3} T^{14} + 33203 p^{4} T^{16} - 828 p^{7} T^{18} + 406 p^{8} T^{20} - 28 p^{10} T^{22} + p^{12} T^{24} \) |
| 13 | \( 1 - 72 T^{2} + 2446 T^{4} - 54792 T^{6} + 980463 T^{8} - 15532784 T^{10} + 217305124 T^{12} - 15532784 p^{2} T^{14} + 980463 p^{4} T^{16} - 54792 p^{6} T^{18} + 2446 p^{8} T^{20} - 72 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - 8 T + 82 T^{2} - 32 p T^{3} + 193 p T^{4} - 960 p T^{5} + 73608 T^{6} - 960 p^{2} T^{7} + 193 p^{3} T^{8} - 32 p^{4} T^{9} + 82 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 160 T^{2} + 11778 T^{4} - 534704 T^{6} + 17109267 T^{8} - 423470864 T^{10} + 8702489988 T^{12} - 423470864 p^{2} T^{14} + 17109267 p^{4} T^{16} - 534704 p^{6} T^{18} + 11778 p^{8} T^{20} - 160 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 - 96 T^{2} + 4750 T^{4} - 133568 T^{6} + 1875743 T^{8} + 10251552 T^{10} - 844938012 T^{12} + 10251552 p^{2} T^{14} + 1875743 p^{4} T^{16} - 133568 p^{6} T^{18} + 4750 p^{8} T^{20} - 96 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( ( 1 - 8 T + 90 T^{2} - 536 T^{3} + 3579 T^{4} - 15808 T^{5} + 104468 T^{6} - 15808 p T^{7} + 3579 p^{2} T^{8} - 536 p^{3} T^{9} + 90 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( ( 1 + 110 T^{2} + 176 T^{3} + 6097 T^{4} + 12072 T^{5} + 229832 T^{6} + 12072 p T^{7} + 6097 p^{2} T^{8} + 176 p^{3} T^{9} + 110 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 12 T + 186 T^{2} - 1612 T^{3} + 14903 T^{4} - 103768 T^{5} + 705548 T^{6} - 103768 p T^{7} + 14903 p^{2} T^{8} - 1612 p^{3} T^{9} + 186 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 8 T + 170 T^{2} - 1016 T^{3} + 11857 T^{4} - 58280 T^{5} + 538520 T^{6} - 58280 p T^{7} + 11857 p^{2} T^{8} - 1016 p^{3} T^{9} + 170 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( 1 - 304 T^{2} + 45438 T^{4} - 4503760 T^{6} + 334179215 T^{8} - 19640799360 T^{10} + 935526889796 T^{12} - 19640799360 p^{2} T^{14} + 334179215 p^{4} T^{16} - 4503760 p^{6} T^{18} + 45438 p^{8} T^{20} - 304 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 - 208 T^{2} + 23730 T^{4} - 1736992 T^{6} + 92911907 T^{8} - 3970546224 T^{10} + 173007979940 T^{12} - 3970546224 p^{2} T^{14} + 92911907 p^{4} T^{16} - 1736992 p^{6} T^{18} + 23730 p^{8} T^{20} - 208 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - 284 T^{2} + 38962 T^{4} - 3563596 T^{6} + 254435519 T^{8} - 15581164472 T^{10} + 861609176572 T^{12} - 15581164472 p^{2} T^{14} + 254435519 p^{4} T^{16} - 3563596 p^{6} T^{18} + 38962 p^{8} T^{20} - 284 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 - 356 T^{2} + 63410 T^{4} - 7697076 T^{6} + 726421631 T^{8} - 56364788040 T^{10} + 3643614187516 T^{12} - 56364788040 p^{2} T^{14} + 726421631 p^{4} T^{16} - 7697076 p^{6} T^{18} + 63410 p^{8} T^{20} - 356 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 - 520 T^{2} + 132302 T^{4} - 21722952 T^{6} + 2561207855 T^{8} - 228277659888 T^{10} + 15762618100900 T^{12} - 228277659888 p^{2} T^{14} + 2561207855 p^{4} T^{16} - 21722952 p^{6} T^{18} + 132302 p^{8} T^{20} - 520 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( ( 1 + 24 T + 518 T^{2} + 6904 T^{3} + 85563 T^{4} + 807232 T^{5} + 7373612 T^{6} + 807232 p T^{7} + 85563 p^{2} T^{8} + 6904 p^{3} T^{9} + 518 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 - 608 T^{2} + 177742 T^{4} - 33204544 T^{6} + 4445090783 T^{8} - 452459122016 T^{10} + 36106645977700 T^{12} - 452459122016 p^{2} T^{14} + 4445090783 p^{4} T^{16} - 33204544 p^{6} T^{18} + 177742 p^{8} T^{20} - 608 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 380 T^{2} + 73958 T^{4} - 10264908 T^{6} + 1143401795 T^{8} - 106217459784 T^{10} + 8380146839308 T^{12} - 106217459784 p^{2} T^{14} + 1143401795 p^{4} T^{16} - 10264908 p^{6} T^{18} + 73958 p^{8} T^{20} - 380 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( 1 - 524 T^{2} + 119722 T^{4} - 15586364 T^{6} + 1277890271 T^{8} - 73296767224 T^{10} + 4411865491692 T^{12} - 73296767224 p^{2} T^{14} + 1277890271 p^{4} T^{16} - 15586364 p^{6} T^{18} + 119722 p^{8} T^{20} - 524 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 8 T + 238 T^{2} - 736 T^{3} + 16265 T^{4} + 47056 T^{5} + 659256 T^{6} + 47056 p T^{7} + 16265 p^{2} T^{8} - 736 p^{3} T^{9} + 238 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 280 T^{2} + 46846 T^{4} - 6268504 T^{6} + 748624287 T^{8} - 77710657552 T^{10} + 7199133986436 T^{12} - 77710657552 p^{2} T^{14} + 748624287 p^{4} T^{16} - 6268504 p^{6} T^{18} + 46846 p^{8} T^{20} - 280 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( ( 1 - 24 T + 574 T^{2} - 9080 T^{3} + 135071 T^{4} - 1580400 T^{5} + 17114692 T^{6} - 1580400 p T^{7} + 135071 p^{2} T^{8} - 9080 p^{3} T^{9} + 574 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−3.06732344679071764483449199609, −2.90270719894242988998370814394, −2.84725552392348183327291162514, −2.84104035902478166441602781566, −2.84074622584097552354550377142, −2.83560836701747833460727565503, −2.62030152398726336488164943097, −2.60408444615964044810854952154, −2.54832265283199589329117170730, −2.37422392260865069614662827825, −2.26775582197195611266641915576, −2.06078779722428732377300390664, −2.02783044405999571758107709594, −1.99729119947617918379566184533, −1.98180194817367139699403886542, −1.59078064153918742971068898342, −1.47679540564015785221400563905, −1.36423336784502729960154062923, −1.32204517816724340311689524994, −1.11177770434776447274125057543, −1.01222236290596235871014741772, −0.969770548717056794941636841558, −0.830385289769384061150848008058, −0.70006251466714807309504223689, −0.60204228706621297474866295032, 0.60204228706621297474866295032, 0.70006251466714807309504223689, 0.830385289769384061150848008058, 0.969770548717056794941636841558, 1.01222236290596235871014741772, 1.11177770434776447274125057543, 1.32204517816724340311689524994, 1.36423336784502729960154062923, 1.47679540564015785221400563905, 1.59078064153918742971068898342, 1.98180194817367139699403886542, 1.99729119947617918379566184533, 2.02783044405999571758107709594, 2.06078779722428732377300390664, 2.26775582197195611266641915576, 2.37422392260865069614662827825, 2.54832265283199589329117170730, 2.60408444615964044810854952154, 2.62030152398726336488164943097, 2.83560836701747833460727565503, 2.84074622584097552354550377142, 2.84104035902478166441602781566, 2.84725552392348183327291162514, 2.90270719894242988998370814394, 3.06732344679071764483449199609
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.