Dirichlet series
| L(s) = 1 | − 4·13-s − 20·17-s + 8·19-s − 2·25-s − 20·37-s + 16·41-s + 16·43-s + 40·47-s + 4·53-s + 16·61-s − 8·67-s + 4·73-s − 16·79-s − 3·81-s − 40·83-s − 52·97-s − 8·101-s + 64·103-s − 48·107-s + 44·113-s + 76·121-s − 16·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 1.10·13-s − 4.85·17-s + 1.83·19-s − 2/5·25-s − 3.28·37-s + 2.49·41-s + 2.43·43-s + 5.83·47-s + 0.549·53-s + 2.04·61-s − 0.977·67-s + 0.468·73-s − 1.80·79-s − 1/3·81-s − 4.39·83-s − 5.27·97-s − 0.796·101-s + 6.30·103-s − 4.64·107-s + 4.13·113-s + 6.90·121-s − 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{84} \cdot 3^{12} \cdot 5^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.68632\times 10^{14}\) |
| Root analytic conductor: | \(3.91551\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{84} \cdot 3^{12} \cdot 5^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4453313111\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4453313111\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{4} )^{3} \) | |
| 5 | \( 1 + 2 T^{2} + 16 T^{3} + 11 T^{4} + 16 T^{5} + 196 T^{6} + 16 p T^{7} + 11 p^{2} T^{8} + 16 p^{3} T^{9} + 2 p^{4} T^{10} + p^{6} T^{12} \) | |
| good | 7 | \( 1 - 32 T^{3} + 78 T^{4} - 64 T^{5} + 512 T^{6} - 3104 T^{7} + 5151 T^{8} - 1504 p T^{9} + 61440 T^{10} - 154656 T^{11} + 212452 T^{12} - 154656 p T^{13} + 61440 p^{2} T^{14} - 1504 p^{4} T^{15} + 5151 p^{4} T^{16} - 3104 p^{5} T^{17} + 512 p^{6} T^{18} - 64 p^{7} T^{19} + 78 p^{8} T^{20} - 32 p^{9} T^{21} + p^{12} T^{24} \) |
| 11 | \( 1 - 76 T^{2} + 2730 T^{4} - 62012 T^{6} + 1017327 T^{8} - 13342808 T^{10} + 153421612 T^{12} - 13342808 p^{2} T^{14} + 1017327 p^{4} T^{16} - 62012 p^{6} T^{18} + 2730 p^{8} T^{20} - 76 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 + 4 T + 8 T^{2} - 20 T^{3} - 46 T^{4} + 1036 T^{5} + 4712 T^{6} + 15876 T^{7} + 11631 T^{8} + 10888 T^{9} + 386384 T^{10} + 2318168 T^{11} + 13644316 T^{12} + 2318168 p T^{13} + 386384 p^{2} T^{14} + 10888 p^{3} T^{15} + 11631 p^{4} T^{16} + 15876 p^{5} T^{17} + 4712 p^{6} T^{18} + 1036 p^{7} T^{19} - 46 p^{8} T^{20} - 20 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 20 T + 200 T^{2} + 1340 T^{3} + 386 p T^{4} + 24940 T^{5} + 84200 T^{6} + 321380 T^{7} + 1425855 T^{8} + 5330760 T^{9} + 10574800 T^{10} - 20440040 T^{11} - 211262532 T^{12} - 20440040 p T^{13} + 10574800 p^{2} T^{14} + 5330760 p^{3} T^{15} + 1425855 p^{4} T^{16} + 321380 p^{5} T^{17} + 84200 p^{6} T^{18} + 24940 p^{7} T^{19} + 386 p^{9} T^{20} + 1340 p^{9} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( ( 1 - 4 T + 46 T^{2} - 108 T^{3} + 1127 T^{4} - 1496 T^{5} + 19236 T^{6} - 1496 p T^{7} + 1127 p^{2} T^{8} - 108 p^{3} T^{9} + 46 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 192 T^{3} - 762 T^{4} + 192 p T^{5} + 18432 T^{6} + 174720 T^{7} - 622993 T^{8} - 4780416 T^{9} - 18432 p^{2} T^{10} + 761856 p T^{11} + 797598100 T^{12} + 761856 p^{2} T^{13} - 18432 p^{4} T^{14} - 4780416 p^{3} T^{15} - 622993 p^{4} T^{16} + 174720 p^{5} T^{17} + 18432 p^{6} T^{18} + 192 p^{8} T^{19} - 762 p^{8} T^{20} - 192 p^{9} T^{21} + p^{12} T^{24} \) | |
| 29 | \( 1 - 196 T^{2} + 16698 T^{4} - 792116 T^{6} + 21850703 T^{8} - 327674472 T^{10} + 4120504204 T^{12} - 327674472 p^{2} T^{14} + 21850703 p^{4} T^{16} - 792116 p^{6} T^{18} + 16698 p^{8} T^{20} - 196 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( 1 - 116 T^{2} + 5826 T^{4} - 173860 T^{6} + 3585903 T^{8} - 39849832 T^{10} + 87053980 T^{12} - 39849832 p^{2} T^{14} + 3585903 p^{4} T^{16} - 173860 p^{6} T^{18} + 5826 p^{8} T^{20} - 116 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 + 20 T + 200 T^{2} + 1324 T^{3} + 6866 T^{4} + 42124 T^{5} + 345768 T^{6} + 2434260 T^{7} + 10271759 T^{8} + 20261256 T^{9} + 87006160 T^{10} + 1713910712 T^{11} + 15428332188 T^{12} + 1713910712 p T^{13} + 87006160 p^{2} T^{14} + 20261256 p^{3} T^{15} + 10271759 p^{4} T^{16} + 2434260 p^{5} T^{17} + 345768 p^{6} T^{18} + 42124 p^{7} T^{19} + 6866 p^{8} T^{20} + 1324 p^{9} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( ( 1 - 8 T + 62 T^{2} - 392 T^{3} + 3215 T^{4} - 24624 T^{5} + 232836 T^{6} - 24624 p T^{7} + 3215 p^{2} T^{8} - 392 p^{3} T^{9} + 62 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( 1 - 16 T + 128 T^{2} - 1008 T^{3} + 11350 T^{4} - 107664 T^{5} + 777856 T^{6} - 5908976 T^{7} + 49140831 T^{8} - 356756640 T^{9} + 2341430528 T^{10} - 16826870112 T^{11} + 119353011764 T^{12} - 16826870112 p T^{13} + 2341430528 p^{2} T^{14} - 356756640 p^{3} T^{15} + 49140831 p^{4} T^{16} - 5908976 p^{5} T^{17} + 777856 p^{6} T^{18} - 107664 p^{7} T^{19} + 11350 p^{8} T^{20} - 1008 p^{9} T^{21} + 128 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 40 T + 800 T^{2} - 11000 T^{3} + 121670 T^{4} - 1191080 T^{5} + 10807200 T^{6} - 91594680 T^{7} + 723942351 T^{8} - 5406086800 T^{9} + 39054830400 T^{10} - 276699722160 T^{11} + 1919879848084 T^{12} - 276699722160 p T^{13} + 39054830400 p^{2} T^{14} - 5406086800 p^{3} T^{15} + 723942351 p^{4} T^{16} - 91594680 p^{5} T^{17} + 10807200 p^{6} T^{18} - 1191080 p^{7} T^{19} + 121670 p^{8} T^{20} - 11000 p^{9} T^{21} + 800 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 4 T + 8 T^{2} - 140 T^{3} - 670 T^{4} + 15828 T^{5} - 48152 T^{6} + 1557116 T^{7} - 9977585 T^{8} - 58146664 T^{9} + 187412048 T^{10} - 2416508280 T^{11} + 32656948988 T^{12} - 2416508280 p T^{13} + 187412048 p^{2} T^{14} - 58146664 p^{3} T^{15} - 9977585 p^{4} T^{16} + 1557116 p^{5} T^{17} - 48152 p^{6} T^{18} + 15828 p^{7} T^{19} - 670 p^{8} T^{20} - 140 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 + 182 T^{2} + 296 T^{3} + 18275 T^{4} + 744 p T^{5} + 1223980 T^{6} + 744 p^{2} T^{7} + 18275 p^{2} T^{8} + 296 p^{3} T^{9} + 182 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 - 8 T + 206 T^{2} - 1384 T^{3} + 21975 T^{4} - 133616 T^{5} + 1627332 T^{6} - 133616 p T^{7} + 21975 p^{2} T^{8} - 1384 p^{3} T^{9} + 206 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 + 8 T + 32 T^{2} + 216 T^{3} - 1066 T^{4} - 5208 T^{5} + 15776 T^{6} + 416568 T^{7} - 8608705 T^{8} - 108730288 T^{9} - 474006208 T^{10} - 219909520 T^{11} + 70187312820 T^{12} - 219909520 p T^{13} - 474006208 p^{2} T^{14} - 108730288 p^{3} T^{15} - 8608705 p^{4} T^{16} + 416568 p^{5} T^{17} + 15776 p^{6} T^{18} - 5208 p^{7} T^{19} - 1066 p^{8} T^{20} + 216 p^{9} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 516 T^{2} + 135810 T^{4} - 23878580 T^{6} + 3098483215 T^{8} - 311206941960 T^{10} + 24748738400924 T^{12} - 311206941960 p^{2} T^{14} + 3098483215 p^{4} T^{16} - 23878580 p^{6} T^{18} + 135810 p^{8} T^{20} - 516 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 4 T + 8 T^{2} - 492 T^{3} + 12714 T^{4} - 4412 T^{5} + 36968 T^{6} - 3317172 T^{7} + 22896943 T^{8} + 353187608 T^{9} - 424155632 T^{10} + 15608674312 T^{11} - 185882933620 T^{12} + 15608674312 p T^{13} - 424155632 p^{2} T^{14} + 353187608 p^{3} T^{15} + 22896943 p^{4} T^{16} - 3317172 p^{5} T^{17} + 36968 p^{6} T^{18} - 4412 p^{7} T^{19} + 12714 p^{8} T^{20} - 492 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + 8 T + 122 T^{2} + 1128 T^{3} + 15279 T^{4} + 124832 T^{5} + 1242924 T^{6} + 124832 p T^{7} + 15279 p^{2} T^{8} + 1128 p^{3} T^{9} + 122 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 40 T + 800 T^{2} + 13144 T^{3} + 217622 T^{4} + 3205416 T^{5} + 40501408 T^{6} + 490221592 T^{7} + 5817267071 T^{8} + 62973667600 T^{9} + 631954107200 T^{10} + 6265495927024 T^{11} + 59602969535796 T^{12} + 6265495927024 p T^{13} + 631954107200 p^{2} T^{14} + 62973667600 p^{3} T^{15} + 5817267071 p^{4} T^{16} + 490221592 p^{5} T^{17} + 40501408 p^{6} T^{18} + 3205416 p^{7} T^{19} + 217622 p^{8} T^{20} + 13144 p^{9} T^{21} + 800 p^{10} T^{22} + 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 572 T^{2} + 164066 T^{4} - 31734668 T^{6} + 4650120463 T^{8} - 546963634808 T^{10} + 53243295060188 T^{12} - 546963634808 p^{2} T^{14} + 4650120463 p^{4} T^{16} - 31734668 p^{6} T^{18} + 164066 p^{8} T^{20} - 572 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 + 52 T + 1352 T^{2} + 25692 T^{3} + 423690 T^{4} + 6304268 T^{5} + 85032488 T^{6} + 1067179332 T^{7} + 12753395791 T^{8} + 145450839496 T^{9} + 1583306609296 T^{10} + 16554696255320 T^{11} + 166450145486924 T^{12} + 16554696255320 p T^{13} + 1583306609296 p^{2} T^{14} + 145450839496 p^{3} T^{15} + 12753395791 p^{4} T^{16} + 1067179332 p^{5} T^{17} + 85032488 p^{6} T^{18} + 6304268 p^{7} T^{19} + 423690 p^{8} T^{20} + 25692 p^{9} T^{21} + 1352 p^{10} T^{22} + 52 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.91538025170933708711610444056, −2.70812511265462961813796894186, −2.61337912906969084278485728625, −2.50238346324690198487879346679, −2.48530490305961598572425040354, −2.41700142939927316772323608882, −2.32336257740735324188189566143, −2.30714034610805007597850324206, −2.02921110352241140448864420462, −1.96962592709993895776642907346, −1.93312374980325825147432139663, −1.91551740908715931316915576488, −1.85702332122907871432624671719, −1.77506777997159829716945502591, −1.42497909735366177257007982072, −1.36994418328955315113997618615, −1.20151694911084539878388180446, −1.19714134861730560422033335804, −0.922095859208662226436249558647, −0.870999602721488595397886689146, −0.76614834787987349693934757280, −0.60266798399236894872123387790, −0.45055602986774896115266205433, −0.27189525647387514136740241142, −0.05145317471136457727837309695, 0.05145317471136457727837309695, 0.27189525647387514136740241142, 0.45055602986774896115266205433, 0.60266798399236894872123387790, 0.76614834787987349693934757280, 0.870999602721488595397886689146, 0.922095859208662226436249558647, 1.19714134861730560422033335804, 1.20151694911084539878388180446, 1.36994418328955315113997618615, 1.42497909735366177257007982072, 1.77506777997159829716945502591, 1.85702332122907871432624671719, 1.91551740908715931316915576488, 1.93312374980325825147432139663, 1.96962592709993895776642907346, 2.02921110352241140448864420462, 2.30714034610805007597850324206, 2.32336257740735324188189566143, 2.41700142939927316772323608882, 2.48530490305961598572425040354, 2.50238346324690198487879346679, 2.61337912906969084278485728625, 2.70812511265462961813796894186, 2.91538025170933708711610444056
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.