Dirichlet series
| L(s) = 1 | − 4·5-s − 18·9-s + 20·17-s − 98·25-s − 24·29-s − 40·37-s − 32·41-s + 72·45-s + 358·49-s + 184·53-s − 276·61-s − 92·73-s + 189·81-s − 80·85-s + 72·89-s − 280·97-s − 40·101-s + 136·109-s + 360·113-s + 846·121-s + 336·125-s + 127-s + 131-s + 137-s + 139-s + 96·145-s + 149-s + ⋯ |
| L(s) = 1 | − 4/5·5-s − 2·9-s + 1.17·17-s − 3.91·25-s − 0.827·29-s − 1.08·37-s − 0.780·41-s + 8/5·45-s + 7.30·49-s + 3.47·53-s − 4.52·61-s − 1.26·73-s + 7/3·81-s − 0.941·85-s + 0.808·89-s − 2.88·97-s − 0.396·101-s + 1.24·109-s + 3.18·113-s + 6.99·121-s + 2.68·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.662·145-s + 0.00671·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.54572\times 10^{16}\) |
| Root analytic conductor: | \(4.98499\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{12} \cdot 19^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.103023850\) |
| \(L(\frac12)\) | \(\approx\) | \(1.103023850\) |
| \(L(2)\) | 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 | \( ( 1 + p T^{2} )^{6} \) | |
| 19 | \( ( 1 + p T^{2} )^{6} \) | |
| good | 5 | \( ( 1 + 2 T + 11 p T^{2} + 146 T^{3} + 1943 T^{4} + 5084 T^{5} + 59666 T^{6} + 5084 p^{2} T^{7} + 1943 p^{4} T^{8} + 146 p^{6} T^{9} + 11 p^{9} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} )^{2} \) |
| 7 | \( 1 - 358 T^{2} + 62991 T^{4} - 7297814 T^{6} + 625416731 T^{8} - 42022210308 T^{10} + 46538740042 p^{2} T^{12} - 42022210308 p^{4} T^{14} + 625416731 p^{8} T^{16} - 7297814 p^{12} T^{18} + 62991 p^{16} T^{20} - 358 p^{20} T^{22} + p^{24} T^{24} \) | |
| 11 | \( 1 - 846 T^{2} + 328327 T^{4} - 77508094 T^{6} + 12646248539 T^{8} - 1613780628500 T^{10} + 191698074801722 T^{12} - 1613780628500 p^{4} T^{14} + 12646248539 p^{8} T^{16} - 77508094 p^{12} T^{18} + 328327 p^{16} T^{20} - 846 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 + 334 T^{2} - 736 T^{3} + 76127 T^{4} - 155744 T^{5} + 16408388 T^{6} - 155744 p^{2} T^{7} + 76127 p^{4} T^{8} - 736 p^{6} T^{9} + 334 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 17 | \( ( 1 - 10 T + 1227 T^{2} - 5306 T^{3} + 620699 T^{4} - 357372 T^{5} + 202098226 T^{6} - 357372 p^{2} T^{7} + 620699 p^{4} T^{8} - 5306 p^{6} T^{9} + 1227 p^{8} T^{10} - 10 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 23 | \( 1 - 2660 T^{2} + 3690914 T^{4} - 3641230804 T^{6} + 2851262197295 T^{8} - 1861703682032712 T^{10} + 1050705683985196764 T^{12} - 1861703682032712 p^{4} T^{14} + 2851262197295 p^{8} T^{16} - 3641230804 p^{12} T^{18} + 3690914 p^{16} T^{20} - 2660 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 + 12 T + 1378 T^{2} + 21052 T^{3} + 1215455 T^{4} + 2254616 T^{5} + 1195285916 T^{6} + 2254616 p^{2} T^{7} + 1215455 p^{4} T^{8} + 21052 p^{6} T^{9} + 1378 p^{8} T^{10} + 12 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 4988 T^{2} + 13551458 T^{4} - 26209260556 T^{6} + 39514173910895 T^{8} - 48928246836414072 T^{10} + 51068756684691755100 T^{12} - 48928246836414072 p^{4} T^{14} + 39514173910895 p^{8} T^{16} - 26209260556 p^{12} T^{18} + 13551458 p^{16} T^{20} - 4988 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 + 20 T + 5978 T^{2} + 87076 T^{3} + 16215263 T^{4} + 171208680 T^{5} + 27119123244 T^{6} + 171208680 p^{2} T^{7} + 16215263 p^{4} T^{8} + 87076 p^{6} T^{9} + 5978 p^{8} T^{10} + 20 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 16 T + 2766 T^{2} - 7120 T^{3} + 1259999 T^{4} - 176080704 T^{5} - 3924664604 T^{6} - 176080704 p^{2} T^{7} + 1259999 p^{4} T^{8} - 7120 p^{6} T^{9} + 2766 p^{8} T^{10} + 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 43 | \( 1 - 5462 T^{2} + 21916191 T^{4} - 58674739654 T^{6} + 130062047531963 T^{8} - 249511372364489412 T^{10} + \)\(45\!\cdots\!18\)\( T^{12} - 249511372364489412 p^{4} T^{14} + 130062047531963 p^{8} T^{16} - 58674739654 p^{12} T^{18} + 21916191 p^{16} T^{20} - 5462 p^{20} T^{22} + p^{24} T^{24} \) | |
| 47 | \( 1 - 15318 T^{2} + 124095751 T^{4} - 676270633726 T^{6} + 2730093815227259 T^{8} - 8534847381009250988 T^{10} + \)\(21\!\cdots\!18\)\( T^{12} - 8534847381009250988 p^{4} T^{14} + 2730093815227259 p^{8} T^{16} - 676270633726 p^{12} T^{18} + 124095751 p^{16} T^{20} - 15318 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 - 92 T + 15970 T^{2} - 1097228 T^{3} + 108023807 T^{4} - 5720458424 T^{5} + 399298723100 T^{6} - 5720458424 p^{2} T^{7} + 108023807 p^{4} T^{8} - 1097228 p^{6} T^{9} + 15970 p^{8} T^{10} - 92 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 - 24364 T^{2} + 297172770 T^{4} - 2401642327484 T^{6} + 14418089457797135 T^{8} - 68215857642461642328 T^{10} + \)\(26\!\cdots\!76\)\( T^{12} - 68215857642461642328 p^{4} T^{14} + 14418089457797135 p^{8} T^{16} - 2401642327484 p^{12} T^{18} + 297172770 p^{16} T^{20} - 24364 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 + 138 T + 20935 T^{2} + 1956530 T^{3} + 182979995 T^{4} + 12746231380 T^{5} + 888223523066 T^{6} + 12746231380 p^{2} T^{7} + 182979995 p^{4} T^{8} + 1956530 p^{6} T^{9} + 20935 p^{8} T^{10} + 138 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 - 23564 T^{2} + 308489250 T^{4} - 2911785005980 T^{6} + 21361176962877263 T^{8} - \)\(12\!\cdots\!60\)\( T^{10} + \)\(62\!\cdots\!68\)\( T^{12} - \)\(12\!\cdots\!60\)\( p^{4} T^{14} + 21361176962877263 p^{8} T^{16} - 2911785005980 p^{12} T^{18} + 308489250 p^{16} T^{20} - 23564 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( 1 - 16172 T^{2} + 187691714 T^{4} - 1499880068092 T^{6} + 9903529153771055 T^{8} - 55927667813600002392 T^{10} + \)\(29\!\cdots\!48\)\( T^{12} - 55927667813600002392 p^{4} T^{14} + 9903529153771055 p^{8} T^{16} - 1499880068092 p^{12} T^{18} + 187691714 p^{16} T^{20} - 16172 p^{20} T^{22} + p^{24} T^{24} \) | |
| 73 | \( ( 1 + 46 T + 12975 T^{2} + 483830 T^{3} + 129206267 T^{4} + 4017619260 T^{5} + 762334223914 T^{6} + 4017619260 p^{2} T^{7} + 129206267 p^{4} T^{8} + 483830 p^{6} T^{9} + 12975 p^{8} T^{10} + 46 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 - 32724 T^{2} + 568306690 T^{4} - 6848530212868 T^{6} + 64130216834894447 T^{8} - \)\(49\!\cdots\!12\)\( T^{10} + \)\(33\!\cdots\!12\)\( T^{12} - \)\(49\!\cdots\!12\)\( p^{4} T^{14} + 64130216834894447 p^{8} T^{16} - 6848530212868 p^{12} T^{18} + 568306690 p^{16} T^{20} - 32724 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( 1 - 29996 T^{2} + 382723298 T^{4} - 2287349407036 T^{6} + 13072186468885 p T^{8} + 95745461886171181992 T^{10} - \)\(92\!\cdots\!04\)\( T^{12} + 95745461886171181992 p^{4} T^{14} + 13072186468885 p^{9} T^{16} - 2287349407036 p^{12} T^{18} + 382723298 p^{16} T^{20} - 29996 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( ( 1 - 36 T + 26466 T^{2} - 86868 T^{3} + 305701647 T^{4} + 6877352376 T^{5} + 2507847403484 T^{6} + 6877352376 p^{2} T^{7} + 305701647 p^{4} T^{8} - 86868 p^{6} T^{9} + 26466 p^{8} T^{10} - 36 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 140 T + 36850 T^{2} + 4370396 T^{3} + 691718831 T^{4} + 66823959704 T^{5} + 8149441212860 T^{6} + 66823959704 p^{2} T^{7} + 691718831 p^{4} T^{8} + 4370396 p^{6} T^{9} + 36850 p^{8} T^{10} + 140 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
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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
−2.96244587088883265633874726788, −2.94495725565433072348988150682, −2.67739358554235121076786744657, −2.65588432079066709186885748697, −2.60973634340011269962265099968, −2.46351058304388938029139373630, −2.34141267826036171528883070958, −2.33291174661078652053716846486, −2.25742351602936512845272621653, −2.19786196442953176847436298066, −1.99883500792228640388690298436, −1.85137484003770442928563625918, −1.82269314475582439771163167388, −1.51248231067005480994526590280, −1.50413255336077029531626510168, −1.48070053164073000857221920134, −1.21016788345353070526597750555, −1.02469706790909629901809860899, −1.00576168119483502590187115538, −0.795429556969657332008822333899, −0.68944737041078441243343250925, −0.48822593687523562634343428726, −0.31700407845496342518686088013, −0.19871236469925492440779185884, −0.11780461191723210668600467693, 0.11780461191723210668600467693, 0.19871236469925492440779185884, 0.31700407845496342518686088013, 0.48822593687523562634343428726, 0.68944737041078441243343250925, 0.795429556969657332008822333899, 1.00576168119483502590187115538, 1.02469706790909629901809860899, 1.21016788345353070526597750555, 1.48070053164073000857221920134, 1.50413255336077029531626510168, 1.51248231067005480994526590280, 1.82269314475582439771163167388, 1.85137484003770442928563625918, 1.99883500792228640388690298436, 2.19786196442953176847436298066, 2.25742351602936512845272621653, 2.33291174661078652053716846486, 2.34141267826036171528883070958, 2.46351058304388938029139373630, 2.60973634340011269962265099968, 2.65588432079066709186885748697, 2.67739358554235121076786744657, 2.94495725565433072348988150682, 2.96244587088883265633874726788
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.