Dirichlet series
| L(s) = 1 | − 3·3-s + 12·9-s + 2·11-s − 7·13-s + 17-s + 2·23-s − 25·27-s + 29-s + 2·31-s − 6·33-s − 20·37-s + 21·39-s − 7·41-s − 19·43-s + 14·47-s − 38·49-s − 3·51-s − 6·53-s − 5·61-s − 14·67-s − 6·69-s + 8·71-s + 9·73-s + 79-s + 62·81-s + 26·83-s − 3·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 4·9-s + 0.603·11-s − 1.94·13-s + 0.242·17-s + 0.417·23-s − 4.81·27-s + 0.185·29-s + 0.359·31-s − 1.04·33-s − 3.28·37-s + 3.36·39-s − 1.09·41-s − 2.89·43-s + 2.04·47-s − 5.42·49-s − 0.420·51-s − 0.824·53-s − 0.640·61-s − 1.71·67-s − 0.722·69-s + 0.949·71-s + 1.05·73-s + 0.112·79-s + 62/9·81-s + 2.85·83-s − 0.321·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 5^{24} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.48719\times 10^{14}\) |
| Root analytic conductor: | \(3.89507\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 5^{24} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3907030335\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3907030335\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| 19 | \( 1 + 33 T^{2} + 126 T^{3} + 660 T^{4} + 4896 T^{5} + 10519 T^{6} + 4896 p T^{7} + 660 p^{2} T^{8} + 126 p^{3} T^{9} + 33 p^{4} T^{10} + p^{6} T^{12} \) | |
| good | 3 | \( 1 + p T - p T^{2} - 20 T^{3} - 11 T^{4} + 50 T^{5} + 46 T^{6} - 151 T^{7} - 202 T^{8} + 433 T^{9} + 979 T^{10} - 133 p T^{11} - 979 p T^{12} - 133 p^{2} T^{13} + 979 p^{2} T^{14} + 433 p^{3} T^{15} - 202 p^{4} T^{16} - 151 p^{5} T^{17} + 46 p^{6} T^{18} + 50 p^{7} T^{19} - 11 p^{8} T^{20} - 20 p^{9} T^{21} - p^{11} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) |
| 7 | \( ( 1 + 19 T^{2} - 2 T^{3} + 232 T^{4} - 52 T^{5} + 1857 T^{6} - 52 p T^{7} + 232 p^{2} T^{8} - 2 p^{3} T^{9} + 19 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 11 | \( ( 1 - T + 4 p T^{2} - 12 T^{3} + 83 p T^{4} + 50 T^{5} + 12127 T^{6} + 50 p T^{7} + 83 p^{3} T^{8} - 12 p^{3} T^{9} + 4 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 13 | \( 1 + 7 T - 9 T^{2} - 110 T^{3} + 275 T^{4} + 1528 T^{5} - 2342 T^{6} - 4999 T^{7} + 6776 T^{8} + 8579 T^{9} + 974699 T^{10} + 1407455 T^{11} - 10852531 T^{12} + 1407455 p T^{13} + 974699 p^{2} T^{14} + 8579 p^{3} T^{15} + 6776 p^{4} T^{16} - 4999 p^{5} T^{17} - 2342 p^{6} T^{18} + 1528 p^{7} T^{19} + 275 p^{8} T^{20} - 110 p^{9} T^{21} - 9 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - T - 53 T^{2} + 118 T^{3} + 1279 T^{4} - 4612 T^{5} - 18478 T^{6} + 115509 T^{7} + 152072 T^{8} - 1941333 T^{9} + 1146995 T^{10} + 14571015 T^{11} - 50735223 T^{12} + 14571015 p T^{13} + 1146995 p^{2} T^{14} - 1941333 p^{3} T^{15} + 152072 p^{4} T^{16} + 115509 p^{5} T^{17} - 18478 p^{6} T^{18} - 4612 p^{7} T^{19} + 1279 p^{8} T^{20} + 118 p^{9} T^{21} - 53 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 2 T - 63 T^{2} - 118 T^{3} + 2797 T^{4} + 10186 T^{5} - 44454 T^{6} - 519144 T^{7} - 416363 T^{8} + 11825274 T^{9} + 66684109 T^{10} - 141247728 T^{11} - 1949433062 T^{12} - 141247728 p T^{13} + 66684109 p^{2} T^{14} + 11825274 p^{3} T^{15} - 416363 p^{4} T^{16} - 519144 p^{5} T^{17} - 44454 p^{6} T^{18} + 10186 p^{7} T^{19} + 2797 p^{8} T^{20} - 118 p^{9} T^{21} - 63 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - T - 133 T^{2} + 22 T^{3} + 9699 T^{4} + 3004 T^{5} - 497206 T^{6} - 242183 T^{7} + 20054396 T^{8} + 8370159 T^{9} - 687070285 T^{10} - 109955845 T^{11} + 20946410613 T^{12} - 109955845 p T^{13} - 687070285 p^{2} T^{14} + 8370159 p^{3} T^{15} + 20054396 p^{4} T^{16} - 242183 p^{5} T^{17} - 497206 p^{6} T^{18} + 3004 p^{7} T^{19} + 9699 p^{8} T^{20} + 22 p^{9} T^{21} - 133 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 - T + 66 T^{2} - 102 T^{3} + 2415 T^{4} + 256 T^{5} + 84673 T^{6} + 256 p T^{7} + 2415 p^{2} T^{8} - 102 p^{3} T^{9} + 66 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( ( 1 + 10 T + 174 T^{2} + 1570 T^{3} + 14519 T^{4} + 106300 T^{5} + 696420 T^{6} + 106300 p T^{7} + 14519 p^{2} T^{8} + 1570 p^{3} T^{9} + 174 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 7 T - 5 T^{2} - 370 T^{3} - 2513 T^{4} + 3064 T^{5} + 240950 T^{6} + 991917 T^{7} - 3406564 T^{8} - 59560149 T^{9} - 243666829 T^{10} + 30479679 p T^{11} + 23036954025 T^{12} + 30479679 p^{2} T^{13} - 243666829 p^{2} T^{14} - 59560149 p^{3} T^{15} - 3406564 p^{4} T^{16} + 991917 p^{5} T^{17} + 240950 p^{6} T^{18} + 3064 p^{7} T^{19} - 2513 p^{8} T^{20} - 370 p^{9} T^{21} - 5 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 19 T + 123 T^{2} - 498 T^{3} - 13795 T^{4} - 91752 T^{5} - 42774 T^{6} + 4007351 T^{7} + 35948718 T^{8} + 152307789 T^{9} - 95214013 T^{10} - 7441755707 T^{11} - 67821535577 T^{12} - 7441755707 p T^{13} - 95214013 p^{2} T^{14} + 152307789 p^{3} T^{15} + 35948718 p^{4} T^{16} + 4007351 p^{5} T^{17} - 42774 p^{6} T^{18} - 91752 p^{7} T^{19} - 13795 p^{8} T^{20} - 498 p^{9} T^{21} + 123 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 14 T - 49 T^{2} + 954 T^{3} + 4781 T^{4} - 23966 T^{5} - 686346 T^{6} + 2698608 T^{7} + 32325253 T^{8} - 92738106 T^{9} - 1771091357 T^{10} - 561038832 T^{11} + 120476925658 T^{12} - 561038832 p T^{13} - 1771091357 p^{2} T^{14} - 92738106 p^{3} T^{15} + 32325253 p^{4} T^{16} + 2698608 p^{5} T^{17} - 686346 p^{6} T^{18} - 23966 p^{7} T^{19} + 4781 p^{8} T^{20} + 954 p^{9} T^{21} - 49 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 6 T - 133 T^{2} - 1462 T^{3} + 5927 T^{4} + 144836 T^{5} + 165844 T^{6} - 9094892 T^{7} - 44103275 T^{8} + 374500350 T^{9} + 3660711409 T^{10} - 6937675270 T^{11} - 214800629242 T^{12} - 6937675270 p T^{13} + 3660711409 p^{2} T^{14} + 374500350 p^{3} T^{15} - 44103275 p^{4} T^{16} - 9094892 p^{5} T^{17} + 165844 p^{6} T^{18} + 144836 p^{7} T^{19} + 5927 p^{8} T^{20} - 1462 p^{9} T^{21} - 133 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 127 T^{2} - 1244 T^{3} + 7433 T^{4} + 145954 T^{5} + 437902 T^{6} - 129632 p T^{7} - 69668015 T^{8} + 1862016 p T^{9} + 3152493097 T^{10} + 5025727318 T^{11} - 146190605686 T^{12} + 5025727318 p T^{13} + 3152493097 p^{2} T^{14} + 1862016 p^{4} T^{15} - 69668015 p^{4} T^{16} - 129632 p^{6} T^{17} + 437902 p^{6} T^{18} + 145954 p^{7} T^{19} + 7433 p^{8} T^{20} - 1244 p^{9} T^{21} - 127 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 5 T - 91 T^{2} - 1136 T^{3} - 1907 T^{4} + 54840 T^{5} + 1000712 T^{6} + 2905517 T^{7} - 43964828 T^{8} - 597265881 T^{9} - 2520150371 T^{10} + 18942017721 T^{11} + 390388229425 T^{12} + 18942017721 p T^{13} - 2520150371 p^{2} T^{14} - 597265881 p^{3} T^{15} - 43964828 p^{4} T^{16} + 2905517 p^{5} T^{17} + 1000712 p^{6} T^{18} + 54840 p^{7} T^{19} - 1907 p^{8} T^{20} - 1136 p^{9} T^{21} - 91 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 14 T - 202 T^{2} - 2440 T^{3} + 39201 T^{4} + 313008 T^{5} - 4908710 T^{6} - 24628686 T^{7} + 482968354 T^{8} + 1250680174 T^{9} - 39772521282 T^{10} - 36273224984 T^{11} + 2745315035105 T^{12} - 36273224984 p T^{13} - 39772521282 p^{2} T^{14} + 1250680174 p^{3} T^{15} + 482968354 p^{4} T^{16} - 24628686 p^{5} T^{17} - 4908710 p^{6} T^{18} + 313008 p^{7} T^{19} + 39201 p^{8} T^{20} - 2440 p^{9} T^{21} - 202 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 8 T - 265 T^{2} + 2484 T^{3} + 38201 T^{4} - 395270 T^{5} - 3666558 T^{6} + 41313984 T^{7} + 254498269 T^{8} - 2765339280 T^{9} - 13917535313 T^{10} + 81449641086 T^{11} + 835069370962 T^{12} + 81449641086 p T^{13} - 13917535313 p^{2} T^{14} - 2765339280 p^{3} T^{15} + 254498269 p^{4} T^{16} + 41313984 p^{5} T^{17} - 3666558 p^{6} T^{18} - 395270 p^{7} T^{19} + 38201 p^{8} T^{20} + 2484 p^{9} T^{21} - 265 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 9 T - 159 T^{2} - 668 T^{3} + 34653 T^{4} + 179280 T^{5} - 1526476 T^{6} - 44503173 T^{7} - 6690216 T^{8} + 2942287653 T^{9} + 30184789965 T^{10} - 146615954265 T^{11} - 2420694726931 T^{12} - 146615954265 p T^{13} + 30184789965 p^{2} T^{14} + 2942287653 p^{3} T^{15} - 6690216 p^{4} T^{16} - 44503173 p^{5} T^{17} - 1526476 p^{6} T^{18} + 179280 p^{7} T^{19} + 34653 p^{8} T^{20} - 668 p^{9} T^{21} - 159 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - T - 293 T^{2} + 218 T^{3} + 43819 T^{4} - 24626 T^{5} - 4449740 T^{6} + 2517869 T^{7} + 354716614 T^{8} - 206333189 T^{9} - 24523535001 T^{10} + 7288133305 T^{11} + 1782018072191 T^{12} + 7288133305 p T^{13} - 24523535001 p^{2} T^{14} - 206333189 p^{3} T^{15} + 354716614 p^{4} T^{16} + 2517869 p^{5} T^{17} - 4449740 p^{6} T^{18} - 24626 p^{7} T^{19} + 43819 p^{8} T^{20} + 218 p^{9} T^{21} - 293 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 13 T + 376 T^{2} - 3210 T^{3} + 57453 T^{4} - 368860 T^{5} + 5527589 T^{6} - 368860 p T^{7} + 57453 p^{2} T^{8} - 3210 p^{3} T^{9} + 376 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 + 8 T - 311 T^{2} - 3440 T^{3} + 47895 T^{4} + 672700 T^{5} - 4263368 T^{6} - 80025552 T^{7} + 220110785 T^{8} + 5983033760 T^{9} - 4346894385 T^{10} - 205258043588 T^{11} - 150723398098 T^{12} - 205258043588 p T^{13} - 4346894385 p^{2} T^{14} + 5983033760 p^{3} T^{15} + 220110785 p^{4} T^{16} - 80025552 p^{5} T^{17} - 4263368 p^{6} T^{18} + 672700 p^{7} T^{19} + 47895 p^{8} T^{20} - 3440 p^{9} T^{21} - 311 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 11 T - 295 T^{2} + 2292 T^{3} + 61369 T^{4} - 292312 T^{5} - 7662036 T^{6} + 14475493 T^{7} + 661057472 T^{8} - 69515193 T^{9} - 32404997495 T^{10} - 37950140887 T^{11} + 1723750419853 T^{12} - 37950140887 p T^{13} - 32404997495 p^{2} T^{14} - 69515193 p^{3} T^{15} + 661057472 p^{4} T^{16} + 14475493 p^{5} T^{17} - 7662036 p^{6} T^{18} - 292312 p^{7} T^{19} + 61369 p^{8} T^{20} + 2292 p^{9} T^{21} - 295 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.85563755281691213441415062341, −2.82263043872671973017879405555, −2.68486528679323345912179829383, −2.58523639293903226815733397841, −2.51240754193919826706642060270, −2.33770681158590106169457694659, −2.15606230017571546897625907907, −2.13052669988068666037307487394, −1.98525737789912676768419921692, −1.96074520178096000600771186959, −1.77451602779589700407949055557, −1.76699253950554310844072609479, −1.74565516648966075403719322804, −1.67681808928030511402484139331, −1.57438907674615184804844472783, −1.28474149138904556803795616018, −1.21805961167911897570330692088, −1.17855288151161613670438961628, −1.11844917775460905460078765419, −0.934536528265668248359509595824, −0.72187537086923257750116386788, −0.64093800462977009161519256921, −0.28308233325644593303004767071, −0.18743638658283572125295718086, −0.11206887802578637142619545873, 0.11206887802578637142619545873, 0.18743638658283572125295718086, 0.28308233325644593303004767071, 0.64093800462977009161519256921, 0.72187537086923257750116386788, 0.934536528265668248359509595824, 1.11844917775460905460078765419, 1.17855288151161613670438961628, 1.21805961167911897570330692088, 1.28474149138904556803795616018, 1.57438907674615184804844472783, 1.67681808928030511402484139331, 1.74565516648966075403719322804, 1.76699253950554310844072609479, 1.77451602779589700407949055557, 1.96074520178096000600771186959, 1.98525737789912676768419921692, 2.13052669988068666037307487394, 2.15606230017571546897625907907, 2.33770681158590106169457694659, 2.51240754193919826706642060270, 2.58523639293903226815733397841, 2.68486528679323345912179829383, 2.82263043872671973017879405555, 2.85563755281691213441415062341
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.