Dirichlet series
| L(s) = 1 | − 6·3-s + 3·4-s − 6·7-s + 15·9-s − 12·11-s − 18·12-s + 3·16-s − 6·19-s + 36·21-s + 4·23-s − 14·27-s − 18·28-s + 72·33-s + 45·36-s − 12·37-s − 10·43-s − 36·44-s − 18·48-s + 13·49-s − 16·53-s + 36·57-s + 24·61-s − 90·63-s − 2·64-s + 6·67-s − 24·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 3/2·4-s − 2.26·7-s + 5·9-s − 3.61·11-s − 5.19·12-s + 3/4·16-s − 1.37·19-s + 7.85·21-s + 0.834·23-s − 2.69·27-s − 3.40·28-s + 12.5·33-s + 15/2·36-s − 1.97·37-s − 1.52·43-s − 5.42·44-s − 2.59·48-s + 13/7·49-s − 2.19·53-s + 4.76·57-s + 3.07·61-s − 11.3·63-s − 1/4·64-s + 0.733·67-s − 2.88·69-s + 1.42·71-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
| \(L(1)\) | \(\approx\) | \(0.1312634638\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1312634638\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 3 | \( ( 1 + T + T^{2} )^{6} \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + 10 T^{2} - 60 T^{3} + 68 T^{4} - 120 T^{5} + 3058 T^{6} - 120 p T^{7} + 68 p^{2} T^{8} - 60 p^{3} T^{9} + 10 p^{4} T^{10} + p^{6} T^{12} \) | |
| good | 7 | \( 1 + 6 T + 23 T^{2} + 66 T^{3} + 134 T^{4} + 330 T^{5} + 1017 T^{6} + 3306 T^{7} + 9598 T^{8} + 20886 T^{9} + 50545 T^{10} + 142914 T^{11} + 360340 T^{12} + 142914 p T^{13} + 50545 p^{2} T^{14} + 20886 p^{3} T^{15} + 9598 p^{4} T^{16} + 3306 p^{5} T^{17} + 1017 p^{6} T^{18} + 330 p^{7} T^{19} + 134 p^{8} T^{20} + 66 p^{9} T^{21} + 23 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) |
| 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 | \( 1 - 52 T^{2} + 168 T^{3} + 1387 T^{4} - 8172 T^{5} - 14204 T^{6} + 12960 p T^{7} - 193484 T^{8} - 210612 p T^{9} + 13181960 T^{10} + 27083988 T^{11} - 297971239 T^{12} + 27083988 p T^{13} + 13181960 p^{2} T^{14} - 210612 p^{4} T^{15} - 193484 p^{4} T^{16} + 12960 p^{6} T^{17} - 14204 p^{6} T^{18} - 8172 p^{7} T^{19} + 1387 p^{8} T^{20} + 168 p^{9} T^{21} - 52 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( 1 - 4 T - 90 T^{2} + 352 T^{3} + 4552 T^{4} - 16660 T^{5} - 162852 T^{6} + 550020 T^{7} + 4500400 T^{8} - 511872 p T^{9} - 107840930 T^{10} + 110877780 T^{11} + 2495445778 T^{12} + 110877780 p T^{13} - 107840930 p^{2} T^{14} - 511872 p^{4} T^{15} + 4500400 p^{4} T^{16} + 550020 p^{5} T^{17} - 162852 p^{6} T^{18} - 16660 p^{7} T^{19} + 4552 p^{8} T^{20} + 352 p^{9} T^{21} - 90 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 12 T + 214 T^{2} + 1992 T^{3} + 21080 T^{4} + 170676 T^{5} + 1479756 T^{6} + 11023596 T^{7} + 85591264 T^{8} + 583783224 T^{9} + 4042005038 T^{10} + 25380883284 T^{11} + 160149152290 T^{12} + 25380883284 p T^{13} + 4042005038 p^{2} T^{14} + 583783224 p^{3} T^{15} + 85591264 p^{4} T^{16} + 11023596 p^{5} T^{17} + 1479756 p^{6} T^{18} + 170676 p^{7} T^{19} + 21080 p^{8} T^{20} + 1992 p^{9} T^{21} + 214 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 10 T - 113 T^{2} - 1194 T^{3} + 7994 T^{4} + 1538 p T^{5} - 631739 T^{6} - 2737954 T^{7} + 45501746 T^{8} + 104252930 T^{9} - 2465984127 T^{10} - 2004512370 T^{11} + 111015536244 T^{12} - 2004512370 p T^{13} - 2465984127 p^{2} T^{14} + 104252930 p^{3} T^{15} + 45501746 p^{4} T^{16} - 2737954 p^{5} T^{17} - 631739 p^{6} T^{18} + 1538 p^{8} T^{19} + 7994 p^{8} T^{20} - 1194 p^{9} T^{21} - 113 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 364 T^{2} + 61770 T^{4} - 6520284 T^{6} + 488357599 T^{8} - 28639110136 T^{10} + 1430995301548 T^{12} - 28639110136 p^{2} T^{14} + 488357599 p^{4} T^{16} - 6520284 p^{6} T^{18} + 61770 p^{8} T^{20} - 364 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 8 T + 244 T^{2} + 1508 T^{3} + 27317 T^{4} + 138500 T^{5} + 1838332 T^{6} + 138500 p T^{7} + 27317 p^{2} T^{8} + 1508 p^{3} T^{9} + 244 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( 1 - 6 T + 217 T^{2} - 1230 T^{3} + 24578 T^{4} - 102678 T^{5} + 1808055 T^{6} - 28278 T^{7} + 61705960 T^{8} + 956637954 T^{9} - 2560310491 T^{10} + 112648845402 T^{11} - 391551549464 T^{12} + 112648845402 p T^{13} - 2560310491 p^{2} T^{14} + 956637954 p^{3} T^{15} + 61705960 p^{4} T^{16} - 28278 p^{5} T^{17} + 1808055 p^{6} T^{18} - 102678 p^{7} T^{19} + 24578 p^{8} T^{20} - 1230 p^{9} T^{21} + 217 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 - 418 T^{2} + 76613 T^{4} - 8785926 T^{6} + 841034218 T^{8} - 78918177002 T^{10} + 6473133473677 T^{12} - 78918177002 p^{2} T^{14} + 841034218 p^{4} T^{16} - 8785926 p^{6} T^{18} + 76613 p^{8} T^{20} - 418 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( 1 - 184 T^{2} + 30000 T^{4} - 2934684 T^{6} + 260990320 T^{8} - 16962802168 T^{10} + 1424504237974 T^{12} - 16962802168 p^{2} T^{14} + 260990320 p^{4} T^{16} - 2934684 p^{6} T^{18} + 30000 p^{8} T^{20} - 184 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( 1 - 12 T + 448 T^{2} - 4800 T^{3} + 103640 T^{4} - 879372 T^{5} + 13901772 T^{6} - 80902836 T^{7} + 1048184536 T^{8} - 856515840 T^{9} + 24224571920 T^{10} + 649972314444 T^{11} - 1767760151450 T^{12} + 649972314444 p T^{13} + 24224571920 p^{2} T^{14} - 856515840 p^{3} T^{15} + 1048184536 p^{4} T^{16} - 80902836 p^{5} T^{17} + 13901772 p^{6} T^{18} - 879372 p^{7} T^{19} + 103640 p^{8} T^{20} - 4800 p^{9} T^{21} + 448 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.91901482205567691476011270503, −2.55828518564514340413343089298, −2.55681596981409516204620911669, −2.51860101231991470110501319176, −2.50311130723620241757707488666, −2.38814620319402301180362807248, −2.35931347547028454489767974052, −2.26785443255854547174115553494, −2.16982104585830832287519464383, −1.98362692053241718958488180981, −1.91891307500332496914326362885, −1.84557625180891160849281380875, −1.72791385987071500488502959019, −1.58535405389491813289255506671, −1.53475882109935599478377733146, −1.29270606902918926373492868929, −1.25994722244204688694241825493, −0.861347921490046392479147579647, −0.837322630507555505173873403674, −0.791243778620755397797294788379, −0.68725479704039656121289944768, −0.57043145794947072796007676526, −0.21747796652001251320413636126, −0.19415694565586654735418572341, −0.15419691308427212412635108950, 0.15419691308427212412635108950, 0.19415694565586654735418572341, 0.21747796652001251320413636126, 0.57043145794947072796007676526, 0.68725479704039656121289944768, 0.791243778620755397797294788379, 0.837322630507555505173873403674, 0.861347921490046392479147579647, 1.25994722244204688694241825493, 1.29270606902918926373492868929, 1.53475882109935599478377733146, 1.58535405389491813289255506671, 1.72791385987071500488502959019, 1.84557625180891160849281380875, 1.91891307500332496914326362885, 1.98362692053241718958488180981, 2.16982104585830832287519464383, 2.26785443255854547174115553494, 2.35931347547028454489767974052, 2.38814620319402301180362807248, 2.50311130723620241757707488666, 2.51860101231991470110501319176, 2.55681596981409516204620911669, 2.55828518564514340413343089298, 2.91901482205567691476011270503
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.