Dirichlet series
| L(s) = 1 | − 6·2-s − 6·3-s + 18·4-s − 6·5-s + 36·6-s − 2·7-s − 36·8-s + 18·9-s + 36·10-s − 108·12-s − 2·13-s + 12·14-s + 36·15-s + 50·16-s − 108·18-s − 108·20-s + 12·21-s + 18·23-s + 216·24-s + 20·25-s + 12·26-s − 30·27-s − 36·28-s − 216·30-s − 4·31-s − 30·32-s + 12·35-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 3.46·3-s + 9·4-s − 2.68·5-s + 14.6·6-s − 0.755·7-s − 12.7·8-s + 6·9-s + 11.3·10-s − 31.1·12-s − 0.554·13-s + 3.20·14-s + 9.29·15-s + 25/2·16-s − 25.4·18-s − 24.1·20-s + 2.61·21-s + 3.75·23-s + 44.0·24-s + 4·25-s + 2.35·26-s − 5.77·27-s − 6.80·28-s − 39.4·30-s − 0.718·31-s − 5.30·32-s + 2.02·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.72385\times 10^{-8}\) |
| Root analytic conductor: | \(0.599438\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.001166422100\) |
| \(L(\frac12)\) | \(\approx\) | \(0.001166422100\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 2 p T + 2 p^{2} T^{2} + 10 p T^{3} + 10 p T^{4} + 4 p^{2} T^{5} + 14 p^{2} T^{6} + 44 p^{2} T^{7} + 91 p^{2} T^{8} + 44 p^{3} T^{9} + 14 p^{4} T^{10} + 4 p^{5} T^{11} + 10 p^{5} T^{12} + 10 p^{6} T^{13} + 2 p^{8} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 6 T + 16 T^{2} + 24 T^{3} + 22 T^{4} + 6 p T^{5} + 136 T^{6} + 126 p T^{7} + 1831 T^{8} + 126 p^{2} T^{9} + 136 p^{2} T^{10} + 6 p^{4} T^{11} + 22 p^{4} T^{12} + 24 p^{5} T^{13} + 16 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 2 | \( 1 + 3 p T + 9 p T^{2} + 9 p^{2} T^{3} + 29 p T^{4} + 39 p T^{5} + 9 p^{3} T^{6} - 149 T^{8} - 177 p T^{9} - 9 p^{6} T^{10} - 183 p^{2} T^{11} - 265 p T^{12} + 15 p^{5} T^{13} + 81 p^{5} T^{14} + 687 p^{3} T^{15} + 8641 T^{16} + 687 p^{4} T^{17} + 81 p^{7} T^{18} + 15 p^{8} T^{19} - 265 p^{5} T^{20} - 183 p^{7} T^{21} - 9 p^{12} T^{22} - 177 p^{8} T^{23} - 149 p^{8} T^{24} + 9 p^{13} T^{26} + 39 p^{12} T^{27} + 29 p^{13} T^{28} + 9 p^{15} T^{29} + 9 p^{15} T^{30} + 3 p^{16} T^{31} + p^{16} T^{32} \) |
| 7 | \( 1 + 2 T + 2 T^{2} - 16 T^{3} - 66 T^{4} - 104 T^{5} + 52 T^{6} + 1072 T^{7} - 235 T^{8} + 2614 T^{9} - 2614 T^{10} + 626 T^{11} + 8430 T^{12} - 2878 p T^{13} + 1536 p^{2} T^{14} - 23844 p^{2} T^{15} + 89248 p^{2} T^{16} - 23844 p^{3} T^{17} + 1536 p^{4} T^{18} - 2878 p^{4} T^{19} + 8430 p^{4} T^{20} + 626 p^{5} T^{21} - 2614 p^{6} T^{22} + 2614 p^{7} T^{23} - 235 p^{8} T^{24} + 1072 p^{9} T^{25} + 52 p^{10} T^{26} - 104 p^{11} T^{27} - 66 p^{12} T^{28} - 16 p^{13} T^{29} + 2 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( ( 1 + 34 T^{2} + 652 T^{4} - 414 T^{5} + 9016 T^{6} - 9072 T^{7} + 100687 T^{8} - 9072 p T^{9} + 9016 p^{2} T^{10} - 414 p^{3} T^{11} + 652 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 + 2 T + 2 T^{2} + 56 T^{3} + 144 T^{4} + 658 T^{5} + 2596 T^{6} + 4150 T^{7} + 35702 T^{8} - 23108 T^{9} - 3274 p T^{10} + 1025978 T^{11} - 7929096 T^{12} - 10433170 T^{13} - 40700658 T^{14} - 417980868 T^{15} - 885935405 T^{16} - 417980868 p T^{17} - 40700658 p^{2} T^{18} - 10433170 p^{3} T^{19} - 7929096 p^{4} T^{20} + 1025978 p^{5} T^{21} - 3274 p^{7} T^{22} - 23108 p^{7} T^{23} + 35702 p^{8} T^{24} + 4150 p^{9} T^{25} + 2596 p^{10} T^{26} + 658 p^{11} T^{27} + 144 p^{12} T^{28} + 56 p^{13} T^{29} + 2 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 964 T^{4} + 237772 T^{8} - 70157828 T^{12} - 47638889354 T^{16} - 70157828 p^{4} T^{20} + 237772 p^{8} T^{24} + 964 p^{12} T^{28} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 92 T^{2} + 4132 T^{4} - 123032 T^{6} + 2693650 T^{8} - 123032 p^{2} T^{10} + 4132 p^{4} T^{12} - 92 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 18 T + 162 T^{2} - 972 T^{3} + 4138 T^{4} - 11184 T^{5} + 3348 T^{6} + 119100 T^{7} - 338219 T^{8} - 2163498 T^{9} + 26656146 T^{10} - 182453766 T^{11} + 892508410 T^{12} - 1883089950 T^{13} - 11227491072 T^{14} + 141260851668 T^{15} - 837597200144 T^{16} + 141260851668 p T^{17} - 11227491072 p^{2} T^{18} - 1883089950 p^{3} T^{19} + 892508410 p^{4} T^{20} - 182453766 p^{5} T^{21} + 26656146 p^{6} T^{22} - 2163498 p^{7} T^{23} - 338219 p^{8} T^{24} + 119100 p^{9} T^{25} + 3348 p^{10} T^{26} - 11184 p^{11} T^{27} + 4138 p^{12} T^{28} - 972 p^{13} T^{29} + 162 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 148 T^{2} + 10806 T^{4} - 562376 T^{6} + 24743345 T^{8} - 973959144 T^{10} + 34914297094 T^{12} - 1155748123996 T^{14} + 35134519733316 T^{16} - 1155748123996 p^{2} T^{18} + 34914297094 p^{4} T^{20} - 973959144 p^{6} T^{22} + 24743345 p^{8} T^{24} - 562376 p^{10} T^{26} + 10806 p^{12} T^{28} - 148 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + 2 T - 78 T^{2} + 76 T^{3} + 3500 T^{4} - 7572 T^{5} - 100340 T^{6} + 131474 T^{7} + 2627523 T^{8} + 131474 p T^{9} - 100340 p^{2} T^{10} - 7572 p^{3} T^{11} + 3500 p^{4} T^{12} + 76 p^{5} T^{13} - 78 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 2 T + 2 T^{2} + 50 T^{3} + 680 T^{4} - 6430 T^{5} + 12750 T^{6} - 67506 T^{7} - 1916078 T^{8} - 67506 p T^{9} + 12750 p^{2} T^{10} - 6430 p^{3} T^{11} + 680 p^{4} T^{12} + 50 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 12 T + 190 T^{2} + 1704 T^{3} + 16861 T^{4} + 130290 T^{5} + 1067482 T^{6} + 7225794 T^{7} + 51027208 T^{8} + 7225794 p T^{9} + 1067482 p^{2} T^{10} + 130290 p^{3} T^{11} + 16861 p^{4} T^{12} + 1704 p^{5} T^{13} + 190 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 + 2 T + 2 T^{2} - 112 T^{3} - 3840 T^{4} - 7982 T^{5} - 2012 T^{6} + 464698 T^{7} + 6098798 T^{8} + 19506388 T^{9} - 1100914 T^{10} - 890327938 T^{11} - 7456003800 T^{12} - 38444523010 T^{13} - 1944910962 T^{14} + 1611174599916 T^{15} + 14665036034491 T^{16} + 1611174599916 p T^{17} - 1944910962 p^{2} T^{18} - 38444523010 p^{3} T^{19} - 7456003800 p^{4} T^{20} - 890327938 p^{5} T^{21} - 1100914 p^{6} T^{22} + 19506388 p^{7} T^{23} + 6098798 p^{8} T^{24} + 464698 p^{9} T^{25} - 2012 p^{10} T^{26} - 7982 p^{11} T^{27} - 3840 p^{12} T^{28} - 112 p^{13} T^{29} + 2 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 12 T + 72 T^{2} + 288 T^{3} - 134 T^{4} - 24762 T^{5} - 246024 T^{6} - 2286006 T^{7} - 18983651 T^{8} - 98916168 T^{9} - 204929766 T^{10} + 1977304884 T^{11} + 21327903874 T^{12} + 124668663210 T^{13} + 894258211950 T^{14} + 7685415723390 T^{15} + 46309407650224 T^{16} + 7685415723390 p T^{17} + 894258211950 p^{2} T^{18} + 124668663210 p^{3} T^{19} + 21327903874 p^{4} T^{20} + 1977304884 p^{5} T^{21} - 204929766 p^{6} T^{22} - 98916168 p^{7} T^{23} - 18983651 p^{8} T^{24} - 2286006 p^{9} T^{25} - 246024 p^{10} T^{26} - 24762 p^{11} T^{27} - 134 p^{12} T^{28} + 288 p^{13} T^{29} + 72 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 6680 T^{4} + 12266524 T^{8} + 25145244376 T^{12} - 161780081477882 T^{16} + 25145244376 p^{4} T^{20} + 12266524 p^{8} T^{24} - 6680 p^{12} T^{28} + p^{16} T^{32} \) | |
| 59 | \( 1 - 328 T^{2} + 54732 T^{4} - 6528848 T^{6} + 640897706 T^{8} - 55004293416 T^{10} + 4215408852016 T^{12} - 289798546717912 T^{14} + 17966936709291123 T^{16} - 289798546717912 p^{2} T^{18} + 4215408852016 p^{4} T^{20} - 55004293416 p^{6} T^{22} + 640897706 p^{8} T^{24} - 6528848 p^{10} T^{26} + 54732 p^{12} T^{28} - 328 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 4 T - 126 T^{2} + 1444 T^{3} + 6365 T^{4} - 126018 T^{5} + 435286 T^{6} + 4572842 T^{7} - 47796264 T^{8} + 4572842 p T^{9} + 435286 p^{2} T^{10} - 126018 p^{3} T^{11} + 6365 p^{4} T^{12} + 1444 p^{5} T^{13} - 126 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 4 T + 8 T^{2} + 1436 T^{3} - 5070 T^{4} + 45274 T^{5} + 890512 T^{6} - 2479154 T^{7} + 72952373 T^{8} + 410747644 T^{9} - 246911326 T^{10} + 51154442144 T^{11} + 220045950810 T^{12} + 1378484916350 T^{13} + 21635958751422 T^{14} + 133914238479582 T^{15} + 1373516219103616 T^{16} + 133914238479582 p T^{17} + 21635958751422 p^{2} T^{18} + 1378484916350 p^{3} T^{19} + 220045950810 p^{4} T^{20} + 51154442144 p^{5} T^{21} - 246911326 p^{6} T^{22} + 410747644 p^{7} T^{23} + 72952373 p^{8} T^{24} - 2479154 p^{9} T^{25} + 890512 p^{10} T^{26} + 45274 p^{11} T^{27} - 5070 p^{12} T^{28} + 1436 p^{13} T^{29} + 8 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 452 T^{2} + 95704 T^{4} - 12356144 T^{6} + 1062541066 T^{8} - 12356144 p^{2} T^{10} + 95704 p^{4} T^{12} - 452 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 4 T + 8 T^{2} - 4 p T^{3} - 844 T^{4} + 28196 T^{5} + 162168 T^{6} - 417348 T^{7} - 41227034 T^{8} - 417348 p T^{9} + 162168 p^{2} T^{10} + 28196 p^{3} T^{11} - 844 p^{4} T^{12} - 4 p^{6} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 + 260 T^{2} + 23724 T^{4} + 1706392 T^{6} + 225017642 T^{8} + 14869625148 T^{10} - 181450880816 T^{12} - 19987311435172 T^{14} + 2160188956785123 T^{16} - 19987311435172 p^{2} T^{18} - 181450880816 p^{4} T^{20} + 14869625148 p^{6} T^{22} + 225017642 p^{8} T^{24} + 1706392 p^{10} T^{26} + 23724 p^{12} T^{28} + 260 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 + 66 T + 2178 T^{2} + 47916 T^{3} + 791830 T^{4} + 10596156 T^{5} + 122712084 T^{6} + 1311937572 T^{7} + 13564902253 T^{8} + 136924099746 T^{9} + 1327585690602 T^{10} + 12258029803398 T^{11} + 109901521003222 T^{12} + 994600086849090 T^{13} + 9293464416138048 T^{14} + 88105947208965780 T^{15} + 817933709355268336 T^{16} + 88105947208965780 p T^{17} + 9293464416138048 p^{2} T^{18} + 994600086849090 p^{3} T^{19} + 109901521003222 p^{4} T^{20} + 12258029803398 p^{5} T^{21} + 1327585690602 p^{6} T^{22} + 136924099746 p^{7} T^{23} + 13564902253 p^{8} T^{24} + 1311937572 p^{9} T^{25} + 122712084 p^{10} T^{26} + 10596156 p^{11} T^{27} + 791830 p^{12} T^{28} + 47916 p^{13} T^{29} + 2178 p^{14} T^{30} + 66 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 412 T^{2} + 69850 T^{4} + 6725680 T^{6} + 546021283 T^{8} + 6725680 p^{2} T^{10} + 69850 p^{4} T^{12} + 412 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 28 T + 392 T^{2} + 872 T^{3} - 117924 T^{4} + 2250436 T^{5} - 16405808 T^{6} - 121301108 T^{7} + 5089087514 T^{8} - 63345173552 T^{9} + 270560580248 T^{10} + 4797083097932 T^{11} - 105427115995440 T^{12} + 964302703147436 T^{13} - 1291090828127400 T^{14} - 85355358908186064 T^{15} + 1288991316725561923 T^{16} - 85355358908186064 p T^{17} - 1291090828127400 p^{2} T^{18} + 964302703147436 p^{3} T^{19} - 105427115995440 p^{4} T^{20} + 4797083097932 p^{5} T^{21} + 270560580248 p^{6} T^{22} - 63345173552 p^{7} T^{23} + 5089087514 p^{8} T^{24} - 121301108 p^{9} T^{25} - 16405808 p^{10} T^{26} + 2250436 p^{11} T^{27} - 117924 p^{12} T^{28} + 872 p^{13} T^{29} + 392 p^{14} T^{30} - 28 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.28376003263830924880713587516, −5.26536851551696707762357532343, −5.26479471397507174895390008046, −5.05079104663487634680836234217, −4.92754752185020384520696212780, −4.86688862183632899891884717176, −4.79838109536486840984948802532, −4.70074455770456811411984524595, −4.54158727975833086853461579263, −4.33850739364639117323882646390, −4.21548998350390319370472016168, −4.16316182600206935947866334557, −4.03074218264799297799391964867, −3.61896249127040887007475587932, −3.55540741718480021542886524616, −3.51034997305632338545401488175, −3.21939218442922471925428814424, −3.20535998862249675067561604912, −2.86093392629714111854334407398, −2.75439607419863582690830338006, −2.66325889852448419303599108902, −2.43624834061761046528913729151, −1.70261660391969818263383751773, −1.59479151241026280775828994449, −1.05222706323018633244862708778, 1.05222706323018633244862708778, 1.59479151241026280775828994449, 1.70261660391969818263383751773, 2.43624834061761046528913729151, 2.66325889852448419303599108902, 2.75439607419863582690830338006, 2.86093392629714111854334407398, 3.20535998862249675067561604912, 3.21939218442922471925428814424, 3.51034997305632338545401488175, 3.55540741718480021542886524616, 3.61896249127040887007475587932, 4.03074218264799297799391964867, 4.16316182600206935947866334557, 4.21548998350390319370472016168, 4.33850739364639117323882646390, 4.54158727975833086853461579263, 4.70074455770456811411984524595, 4.79838109536486840984948802532, 4.86688862183632899891884717176, 4.92754752185020384520696212780, 5.05079104663487634680836234217, 5.26479471397507174895390008046, 5.26536851551696707762357532343, 5.28376003263830924880713587516
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.