Dirichlet series
| L(s) = 1 | − 6·2-s + 15·4-s + 4·7-s − 14·8-s + 3·9-s − 12·11-s − 4·13-s − 24·14-s − 21·16-s − 18·18-s + 6·19-s + 72·22-s − 12·23-s + 24·26-s + 60·28-s + 84·32-s + 45·36-s + 12·37-s − 36·38-s − 12·43-s − 180·44-s + 72·46-s − 16·47-s + 13·49-s − 60·52-s − 56·56-s + 24·61-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 15/2·4-s + 1.51·7-s − 4.94·8-s + 9-s − 3.61·11-s − 1.10·13-s − 6.41·14-s − 5.25·16-s − 4.24·18-s + 1.37·19-s + 15.3·22-s − 2.50·23-s + 4.70·26-s + 11.3·28-s + 14.8·32-s + 15/2·36-s + 1.97·37-s − 5.83·38-s − 1.82·43-s − 27.1·44-s + 10.6·46-s − 2.33·47-s + 13/7·49-s − 8.32·52-s − 7.48·56-s + 3.07·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.03114\times 10^{14}\) |
| Root analytic conductor: | \(3.94598\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.07244034573\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07244034573\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{6} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{3} \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + 4 T - 2 T^{2} + 32 T^{3} + 244 T^{4} + 388 T^{5} + 614 T^{6} + 388 p T^{7} + 244 p^{2} T^{8} + 32 p^{3} T^{9} - 2 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( 1 - 4 T + 3 T^{2} + 12 T^{3} - 50 T^{4} + 80 T^{5} - 39 T^{6} - 600 T^{7} + 2578 T^{8} - 4572 T^{9} - 5011 T^{10} + 71100 T^{11} - 244880 T^{12} + 71100 p T^{13} - 5011 p^{2} T^{14} - 4572 p^{3} T^{15} + 2578 p^{4} T^{16} - 600 p^{5} T^{17} - 39 p^{6} T^{18} + 80 p^{7} T^{19} - 50 p^{8} T^{20} + 12 p^{9} T^{21} + 3 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 + 12 T + 92 T^{2} + 48 p T^{3} + 2520 T^{4} + 10476 T^{5} + 35196 T^{6} + 90636 T^{7} + 116464 T^{8} - 378768 T^{9} - 3813556 T^{10} - 19520724 T^{11} - 72726650 T^{12} - 19520724 p T^{13} - 3813556 p^{2} T^{14} - 378768 p^{3} T^{15} + 116464 p^{4} T^{16} + 90636 p^{5} T^{17} + 35196 p^{6} T^{18} + 10476 p^{7} T^{19} + 2520 p^{8} T^{20} + 48 p^{10} T^{21} + 92 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 52 T^{2} + 1387 T^{4} - 2412 T^{5} + 28316 T^{6} - 118944 T^{7} + 445588 T^{8} - 3008484 T^{9} + 7326376 T^{10} - 62566812 T^{11} + 138037145 T^{12} - 62566812 p T^{13} + 7326376 p^{2} T^{14} - 3008484 p^{3} T^{15} + 445588 p^{4} T^{16} - 118944 p^{5} T^{17} + 28316 p^{6} T^{18} - 2412 p^{7} T^{19} + 1387 p^{8} T^{20} + 52 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 - 6 T + 55 T^{2} - 258 T^{3} + 938 T^{4} - 222 p T^{5} + 12861 T^{6} - 52650 T^{7} + 276010 T^{8} - 297414 T^{9} - 1151839 T^{10} + 15782310 T^{11} - 135483788 T^{12} + 15782310 p T^{13} - 1151839 p^{2} T^{14} - 297414 p^{3} T^{15} + 276010 p^{4} T^{16} - 52650 p^{5} T^{17} + 12861 p^{6} T^{18} - 222 p^{8} T^{19} + 938 p^{8} T^{20} - 258 p^{9} T^{21} + 55 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 12 T + 170 T^{2} + 1464 T^{3} + 13128 T^{4} + 95652 T^{5} + 688836 T^{6} + 4417260 T^{7} + 27349168 T^{8} + 156378984 T^{9} + 859327730 T^{10} + 4425136548 T^{11} + 21885412882 T^{12} + 4425136548 p T^{13} + 859327730 p^{2} T^{14} + 156378984 p^{3} T^{15} + 27349168 p^{4} T^{16} + 4417260 p^{5} T^{17} + 688836 p^{6} T^{18} + 95652 p^{7} T^{19} + 13128 p^{8} T^{20} + 1464 p^{9} T^{21} + 170 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 82 T^{2} - 480 T^{3} + 3253 T^{4} + 33552 T^{5} + 12298 T^{6} - 1039680 T^{7} - 4012670 T^{8} + 11993808 T^{9} + 101280566 T^{10} + 47802384 T^{11} - 2150755819 T^{12} + 47802384 p T^{13} + 101280566 p^{2} T^{14} + 11993808 p^{3} T^{15} - 4012670 p^{4} T^{16} - 1039680 p^{5} T^{17} + 12298 p^{6} T^{18} + 33552 p^{7} T^{19} + 3253 p^{8} T^{20} - 480 p^{9} T^{21} - 82 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( 1 - 160 T^{2} + 14714 T^{4} - 954888 T^{6} + 48159331 T^{8} - 1967505128 T^{10} + 66774795364 T^{12} - 1967505128 p^{2} T^{14} + 48159331 p^{4} T^{16} - 954888 p^{6} T^{18} + 14714 p^{8} T^{20} - 160 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 - 12 T - 70 T^{2} + 984 T^{3} + 5816 T^{4} - 50820 T^{5} - 457644 T^{6} + 2440212 T^{7} + 23935072 T^{8} - 72803928 T^{9} - 1118977118 T^{10} + 832120572 T^{11} + 47421054562 T^{12} + 832120572 p T^{13} - 1118977118 p^{2} T^{14} - 72803928 p^{3} T^{15} + 23935072 p^{4} T^{16} + 2440212 p^{5} T^{17} - 457644 p^{6} T^{18} - 50820 p^{7} T^{19} + 5816 p^{8} T^{20} + 984 p^{9} T^{21} - 70 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 196 T^{2} + 21019 T^{4} + 15372 T^{5} + 1607852 T^{6} + 2315232 T^{7} + 96955252 T^{8} + 186358788 T^{9} + 4941489880 T^{10} + 10348695900 T^{11} + 217634194985 T^{12} + 10348695900 p T^{13} + 4941489880 p^{2} T^{14} + 186358788 p^{3} T^{15} + 96955252 p^{4} T^{16} + 2315232 p^{5} T^{17} + 1607852 p^{6} T^{18} + 15372 p^{7} T^{19} + 21019 p^{8} T^{20} + 196 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 + 12 T + 235 T^{2} + 2244 T^{3} + 25634 T^{4} + 192456 T^{5} + 1712469 T^{6} + 10617264 T^{7} + 83481046 T^{8} + 465064116 T^{9} + 3566391965 T^{10} + 19204758132 T^{11} + 151641605488 T^{12} + 19204758132 p T^{13} + 3566391965 p^{2} T^{14} + 465064116 p^{3} T^{15} + 83481046 p^{4} T^{16} + 10617264 p^{5} T^{17} + 1712469 p^{6} T^{18} + 192456 p^{7} T^{19} + 25634 p^{8} T^{20} + 2244 p^{9} T^{21} + 235 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 + 8 T + 214 T^{2} + 1688 T^{3} + 21491 T^{4} + 149360 T^{5} + 1280620 T^{6} + 149360 p T^{7} + 21491 p^{2} T^{8} + 1688 p^{3} T^{9} + 214 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 8 p T^{2} + 90042 T^{4} - 12517296 T^{6} + 1261627411 T^{8} - 96818608024 T^{10} + 5791624913956 T^{12} - 96818608024 p^{2} T^{14} + 1261627411 p^{4} T^{16} - 12517296 p^{6} T^{18} + 90042 p^{8} T^{20} - 8 p^{11} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 + 250 T^{2} + 32713 T^{4} - 20232 T^{5} + 3093782 T^{6} - 4441680 T^{7} + 239943118 T^{8} - 507769416 T^{9} + 16484265514 T^{10} - 40270899528 T^{11} + 1025484358349 T^{12} - 40270899528 p T^{13} + 16484265514 p^{2} T^{14} - 507769416 p^{3} T^{15} + 239943118 p^{4} T^{16} - 4441680 p^{5} T^{17} + 3093782 p^{6} T^{18} - 20232 p^{7} T^{19} + 32713 p^{8} T^{20} + 250 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 - 24 T + 38 T^{2} + 1992 T^{3} + 15152 T^{4} - 341688 T^{5} - 1541460 T^{6} + 21800064 T^{7} + 260618440 T^{8} - 1575800472 T^{9} - 19655273522 T^{10} + 16321937280 T^{11} + 1648091128162 T^{12} + 16321937280 p T^{13} - 19655273522 p^{2} T^{14} - 1575800472 p^{3} T^{15} + 260618440 p^{4} T^{16} + 21800064 p^{5} T^{17} - 1541460 p^{6} T^{18} - 341688 p^{7} T^{19} + 15152 p^{8} T^{20} + 1992 p^{9} T^{21} + 38 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 24 T + 89 T^{2} - 2040 T^{3} - 12178 T^{4} + 134520 T^{5} + 624159 T^{6} - 10703976 T^{7} - 31388888 T^{8} + 743618472 T^{9} + 3423470341 T^{10} - 19530347208 T^{11} - 253683006716 T^{12} - 19530347208 p T^{13} + 3423470341 p^{2} T^{14} + 743618472 p^{3} T^{15} - 31388888 p^{4} T^{16} - 10703976 p^{5} T^{17} + 624159 p^{6} T^{18} + 134520 p^{7} T^{19} - 12178 p^{8} T^{20} - 2040 p^{9} T^{21} + 89 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 12 T + 110 T^{2} - 744 T^{3} - 4320 T^{4} + 76692 T^{5} - 1170036 T^{6} + 12919788 T^{7} - 53274920 T^{8} + 143658264 T^{9} + 4093998806 T^{10} - 70223548068 T^{11} + 498448543282 T^{12} - 70223548068 p T^{13} + 4093998806 p^{2} T^{14} + 143658264 p^{3} T^{15} - 53274920 p^{4} T^{16} + 12919788 p^{5} T^{17} - 1170036 p^{6} T^{18} + 76692 p^{7} T^{19} - 4320 p^{8} T^{20} - 744 p^{9} T^{21} + 110 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 - 20 T + 409 T^{2} - 6076 T^{3} + 76186 T^{4} - 792596 T^{5} + 7543697 T^{6} - 792596 p T^{7} + 76186 p^{2} T^{8} - 6076 p^{3} T^{9} + 409 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 26 T + 401 T^{2} + 4446 T^{3} + 40363 T^{4} + 322972 T^{5} + 2678236 T^{6} + 322972 p T^{7} + 40363 p^{2} T^{8} + 4446 p^{3} T^{9} + 401 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( ( 1 + 16 T + 220 T^{2} + 772 T^{3} + 3152 T^{4} - 1120 p T^{5} - 415466 T^{6} - 1120 p^{2} T^{7} + 3152 p^{2} T^{8} + 772 p^{3} T^{9} + 220 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 + 24 T + 640 T^{2} + 10752 T^{3} + 176947 T^{4} + 2389044 T^{5} + 31865720 T^{6} + 376949784 T^{7} + 4462051804 T^{8} + 47946764580 T^{9} + 514484570908 T^{10} + 5069907803652 T^{11} + 49769429455721 T^{12} + 5069907803652 p T^{13} + 514484570908 p^{2} T^{14} + 47946764580 p^{3} T^{15} + 4462051804 p^{4} T^{16} + 376949784 p^{5} T^{17} + 31865720 p^{6} T^{18} + 2389044 p^{7} T^{19} + 176947 p^{8} T^{20} + 10752 p^{9} T^{21} + 640 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 24 T - 88 T^{2} - 4992 T^{3} + 22712 T^{4} + 836616 T^{5} - 3722556 T^{6} - 77720520 T^{7} + 827801608 T^{8} + 5141457024 T^{9} - 130599330536 T^{10} - 181557789144 T^{11} + 14624775033862 T^{12} - 181557789144 p T^{13} - 130599330536 p^{2} T^{14} + 5141457024 p^{3} T^{15} + 827801608 p^{4} T^{16} - 77720520 p^{5} T^{17} - 3722556 p^{6} T^{18} + 836616 p^{7} T^{19} + 22712 p^{8} T^{20} - 4992 p^{9} T^{21} - 88 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.66319147493757764564942638787, −2.60032007812541199572119576699, −2.48968344067262804277056280917, −2.46225890392534059914755603731, −2.31335803366211510782703627839, −2.23572019295249858676719182955, −2.20625056079292941544811885924, −2.11499000083912831545457710924, −2.06598532929937534203898887206, −1.99891547896166752714384581945, −1.96386332955484509218552224669, −1.59226242028706743300046215697, −1.47861248761627478182887580973, −1.44164312313267167284283706627, −1.39810698719252572722011671377, −1.37722934379442543461538216658, −1.37385432602971603789633345165, −1.09218248259832200608705853383, −0.839207806437516299697471023643, −0.789840841739644763751363120838, −0.72270717882046090222917853684, −0.51992960426285209505541033357, −0.25605850603337889980290433134, −0.23193307691663124022822602636, −0.13870271607291597194643740250, 0.13870271607291597194643740250, 0.23193307691663124022822602636, 0.25605850603337889980290433134, 0.51992960426285209505541033357, 0.72270717882046090222917853684, 0.789840841739644763751363120838, 0.839207806437516299697471023643, 1.09218248259832200608705853383, 1.37385432602971603789633345165, 1.37722934379442543461538216658, 1.39810698719252572722011671377, 1.44164312313267167284283706627, 1.47861248761627478182887580973, 1.59226242028706743300046215697, 1.96386332955484509218552224669, 1.99891547896166752714384581945, 2.06598532929937534203898887206, 2.11499000083912831545457710924, 2.20625056079292941544811885924, 2.23572019295249858676719182955, 2.31335803366211510782703627839, 2.46225890392534059914755603731, 2.48968344067262804277056280917, 2.60032007812541199572119576699, 2.66319147493757764564942638787
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.