Dirichlet series
| L(s) = 1 | + 40·5-s − 11·7-s − 29·9-s − 41·11-s + 28·13-s + 71·17-s − 177·19-s − 184·23-s + 900·25-s + 165·27-s + 225·29-s + 36·31-s − 440·35-s − 348·37-s + 620·41-s + 390·43-s − 1.16e3·45-s − 123·47-s − 871·49-s + 1.40e3·53-s − 1.64e3·55-s + 676·59-s + 1.44e3·61-s + 319·63-s + 1.12e3·65-s + 1.58e3·67-s − 1.39e3·71-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 0.593·7-s − 1.07·9-s − 1.12·11-s + 0.597·13-s + 1.01·17-s − 2.13·19-s − 1.66·23-s + 36/5·25-s + 1.17·27-s + 1.44·29-s + 0.208·31-s − 2.12·35-s − 1.54·37-s + 2.36·41-s + 1.38·43-s − 3.84·45-s − 0.381·47-s − 2.53·49-s + 3.64·53-s − 4.02·55-s + 1.49·59-s + 3.03·61-s + 0.637·63-s + 2.13·65-s + 2.88·67-s − 2.33·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 5^{8} \cdot 23^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.92961\times 10^{16}\) |
| Root analytic conductor: | \(10.4193\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 5^{8} \cdot 23^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(126.2154696\) |
| \(L(\frac12)\) | \(\approx\) | \(126.2154696\) |
| \(L(\frac{5}{2})\) | 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 | \( ( 1 - p T )^{8} \) | |
| 23 | \( ( 1 + p T )^{8} \) | |
| good | 3 | \( 1 + 29 T^{2} - 55 p T^{3} + 136 p T^{4} - 2278 p T^{5} + 31583 T^{6} - 47803 p T^{7} + 1192870 T^{8} - 47803 p^{4} T^{9} + 31583 p^{6} T^{10} - 2278 p^{10} T^{11} + 136 p^{13} T^{12} - 55 p^{16} T^{13} + 29 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( 1 + 11 T + 992 T^{2} + 3872 T^{3} + 472349 T^{4} + 177192 p T^{5} + 166603296 T^{6} - 29395885 T^{7} + 46236483812 T^{8} - 29395885 p^{3} T^{9} + 166603296 p^{6} T^{10} + 177192 p^{10} T^{11} + 472349 p^{12} T^{12} + 3872 p^{15} T^{13} + 992 p^{18} T^{14} + 11 p^{21} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 + 41 T + 5873 T^{2} + 157005 T^{3} + 14068482 T^{4} + 180135085 T^{5} + 18466017895 T^{6} - 16339547375 T^{7} + 20718442622650 T^{8} - 16339547375 p^{3} T^{9} + 18466017895 p^{6} T^{10} + 180135085 p^{9} T^{11} + 14068482 p^{12} T^{12} + 157005 p^{15} T^{13} + 5873 p^{18} T^{14} + 41 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 28 T + 10183 T^{2} - 186629 T^{3} + 53581732 T^{4} - 799144832 T^{5} + 190756469999 T^{6} - 2316286663399 T^{7} + 485049497521610 T^{8} - 2316286663399 p^{3} T^{9} + 190756469999 p^{6} T^{10} - 799144832 p^{9} T^{11} + 53581732 p^{12} T^{12} - 186629 p^{15} T^{13} + 10183 p^{18} T^{14} - 28 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 71 T + 18550 T^{2} - 49654 p T^{3} + 162807583 T^{4} - 4880541902 T^{5} + 994551629302 T^{6} - 21624905990685 T^{7} + 5200106445915736 T^{8} - 21624905990685 p^{3} T^{9} + 994551629302 p^{6} T^{10} - 4880541902 p^{9} T^{11} + 162807583 p^{12} T^{12} - 49654 p^{16} T^{13} + 18550 p^{18} T^{14} - 71 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 177 T + 39187 T^{2} + 4065213 T^{3} + 454553614 T^{4} + 25459408197 T^{5} + 1353294771973 T^{6} - 31423829302863 T^{7} - 3396329842929950 T^{8} - 31423829302863 p^{3} T^{9} + 1353294771973 p^{6} T^{10} + 25459408197 p^{9} T^{11} + 454553614 p^{12} T^{12} + 4065213 p^{15} T^{13} + 39187 p^{18} T^{14} + 177 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 - 225 T + 108206 T^{2} - 19791225 T^{3} + 5625653433 T^{4} - 955817479364 T^{5} + 207290589945806 T^{6} - 32990624027850378 T^{7} + 5863159759361827628 T^{8} - 32990624027850378 p^{3} T^{9} + 207290589945806 p^{6} T^{10} - 955817479364 p^{9} T^{11} + 5625653433 p^{12} T^{12} - 19791225 p^{15} T^{13} + 108206 p^{18} T^{14} - 225 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 - 36 T + 128202 T^{2} - 1766971 T^{3} + 8846760489 T^{4} - 59582646671 T^{5} + 415042259540374 T^{6} - 1208520514388535 T^{7} + 14237380713790649612 T^{8} - 1208520514388535 p^{3} T^{9} + 415042259540374 p^{6} T^{10} - 59582646671 p^{9} T^{11} + 8846760489 p^{12} T^{12} - 1766971 p^{15} T^{13} + 128202 p^{18} T^{14} - 36 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 + 348 T + 329045 T^{2} + 102878626 T^{3} + 51137299426 T^{4} + 13843050752210 T^{5} + 4810260448068899 T^{6} + 1100796051798899704 T^{7} + \)\(29\!\cdots\!30\)\( T^{8} + 1100796051798899704 p^{3} T^{9} + 4810260448068899 p^{6} T^{10} + 13843050752210 p^{9} T^{11} + 51137299426 p^{12} T^{12} + 102878626 p^{15} T^{13} + 329045 p^{18} T^{14} + 348 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 620 T + 500094 T^{2} - 224816633 T^{3} + 106454294243 T^{4} - 38505414842223 T^{5} + 13487249042978034 T^{6} - 4032154543722725049 T^{7} + \)\(11\!\cdots\!86\)\( T^{8} - 4032154543722725049 p^{3} T^{9} + 13487249042978034 p^{6} T^{10} - 38505414842223 p^{9} T^{11} + 106454294243 p^{12} T^{12} - 224816633 p^{15} T^{13} + 500094 p^{18} T^{14} - 620 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 390 T + 305460 T^{2} - 103396782 T^{3} + 53816476148 T^{4} - 16102782377502 T^{5} + 6599877264299052 T^{6} - 1722769588244216742 T^{7} + \)\(60\!\cdots\!82\)\( T^{8} - 1722769588244216742 p^{3} T^{9} + 6599877264299052 p^{6} T^{10} - 16102782377502 p^{9} T^{11} + 53816476148 p^{12} T^{12} - 103396782 p^{15} T^{13} + 305460 p^{18} T^{14} - 390 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 + 123 T + 11431 p T^{2} + 47432616 T^{3} + 134846786232 T^{4} + 7925188937146 T^{5} + 21608963378329919 T^{6} + 859204272702381501 T^{7} + \)\(25\!\cdots\!94\)\( T^{8} + 859204272702381501 p^{3} T^{9} + 21608963378329919 p^{6} T^{10} + 7925188937146 p^{9} T^{11} + 134846786232 p^{12} T^{12} + 47432616 p^{15} T^{13} + 11431 p^{19} T^{14} + 123 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 1406 T + 1450549 T^{2} - 1131248566 T^{3} + 750956817646 T^{4} - 430416975525618 T^{5} + 217485756524875507 T^{6} - 98584836535784694842 T^{7} + \)\(39\!\cdots\!06\)\( T^{8} - 98584836535784694842 p^{3} T^{9} + 217485756524875507 p^{6} T^{10} - 430416975525618 p^{9} T^{11} + 750956817646 p^{12} T^{12} - 1131248566 p^{15} T^{13} + 1450549 p^{18} T^{14} - 1406 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 676 T + 945729 T^{2} - 9229250 p T^{3} + 494796808106 T^{4} - 241047732663330 T^{5} + 165074390540687703 T^{6} - 69870881622242788296 T^{7} + \)\(39\!\cdots\!82\)\( T^{8} - 69870881622242788296 p^{3} T^{9} + 165074390540687703 p^{6} T^{10} - 241047732663330 p^{9} T^{11} + 494796808106 p^{12} T^{12} - 9229250 p^{16} T^{13} + 945729 p^{18} T^{14} - 676 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 1447 T + 2371267 T^{2} - 2222594555 T^{3} + 2120966819038 T^{4} - 1471195013682537 T^{5} + 1014315904689582453 T^{6} - \)\(54\!\cdots\!25\)\( T^{7} + \)\(29\!\cdots\!70\)\( T^{8} - \)\(54\!\cdots\!25\)\( p^{3} T^{9} + 1014315904689582453 p^{6} T^{10} - 1471195013682537 p^{9} T^{11} + 2120966819038 p^{12} T^{12} - 2222594555 p^{15} T^{13} + 2371267 p^{18} T^{14} - 1447 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 - 1582 T + 2671913 T^{2} - 2885199986 T^{3} + 2910295279454 T^{4} - 2379458358971522 T^{5} + 1767771704076688047 T^{6} - \)\(11\!\cdots\!86\)\( T^{7} + \)\(66\!\cdots\!90\)\( T^{8} - \)\(11\!\cdots\!86\)\( p^{3} T^{9} + 1767771704076688047 p^{6} T^{10} - 2379458358971522 p^{9} T^{11} + 2910295279454 p^{12} T^{12} - 2885199986 p^{15} T^{13} + 2671913 p^{18} T^{14} - 1582 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 + 1396 T + 2335660 T^{2} + 1693070851 T^{3} + 1445058509101 T^{4} + 393020845767379 T^{5} + 147829217354453770 T^{6} - \)\(20\!\cdots\!21\)\( T^{7} - \)\(82\!\cdots\!44\)\( T^{8} - \)\(20\!\cdots\!21\)\( p^{3} T^{9} + 147829217354453770 p^{6} T^{10} + 393020845767379 p^{9} T^{11} + 1445058509101 p^{12} T^{12} + 1693070851 p^{15} T^{13} + 2335660 p^{18} T^{14} + 1396 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 17 T + 1310239 T^{2} + 111342306 T^{3} + 1036258807762 T^{4} + 24516597356860 T^{5} + 593265408885895961 T^{6} + 7924080723309964795 T^{7} + \)\(25\!\cdots\!94\)\( T^{8} + 7924080723309964795 p^{3} T^{9} + 593265408885895961 p^{6} T^{10} + 24516597356860 p^{9} T^{11} + 1036258807762 p^{12} T^{12} + 111342306 p^{15} T^{13} + 1310239 p^{18} T^{14} - 17 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 + 708 T + 3375072 T^{2} + 2123461964 T^{3} + 5232562600812 T^{4} + 2856139766234764 T^{5} + 4848938155595858272 T^{6} + \)\(22\!\cdots\!84\)\( T^{7} + \)\(29\!\cdots\!06\)\( T^{8} + \)\(22\!\cdots\!84\)\( p^{3} T^{9} + 4848938155595858272 p^{6} T^{10} + 2856139766234764 p^{9} T^{11} + 5232562600812 p^{12} T^{12} + 2123461964 p^{15} T^{13} + 3375072 p^{18} T^{14} + 708 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 + 1486 T + 3175749 T^{2} + 2764279966 T^{3} + 3827028296642 T^{4} + 2661458902467278 T^{5} + 3344306091213417603 T^{6} + \)\(21\!\cdots\!78\)\( T^{7} + \)\(22\!\cdots\!46\)\( T^{8} + \)\(21\!\cdots\!78\)\( p^{3} T^{9} + 3344306091213417603 p^{6} T^{10} + 2661458902467278 p^{9} T^{11} + 3827028296642 p^{12} T^{12} + 2764279966 p^{15} T^{13} + 3175749 p^{18} T^{14} + 1486 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 - 1360 T + 5306992 T^{2} - 5889125224 T^{3} + 12430790952652 T^{4} - 11390623252430824 T^{5} + 16927591488285602128 T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!54\)\( T^{8} - \)\(12\!\cdots\!60\)\( p^{3} T^{9} + 16927591488285602128 p^{6} T^{10} - 11390623252430824 p^{9} T^{11} + 12430790952652 p^{12} T^{12} - 5889125224 p^{15} T^{13} + 5306992 p^{18} T^{14} - 1360 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 + 855 T + 2246661 T^{2} + 1636833709 T^{3} + 3720417362430 T^{4} + 2126288854025687 T^{5} + 3672588527742938779 T^{6} + \)\(20\!\cdots\!45\)\( T^{7} + \)\(37\!\cdots\!30\)\( T^{8} + \)\(20\!\cdots\!45\)\( p^{3} T^{9} + 3672588527742938779 p^{6} T^{10} + 2126288854025687 p^{9} T^{11} + 3720417362430 p^{12} T^{12} + 1636833709 p^{15} T^{13} + 2246661 p^{18} T^{14} + 855 p^{21} T^{15} + p^{24} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.60795758996711955335859338440, −3.17490984374461120607307955162, −2.96557872015502756373934619578, −2.96003783906370937210071995691, −2.95622444013472081082784267488, −2.76763016875216247281196840499, −2.74801252452625671302720340078, −2.65850305039670375424294482375, −2.50499634956667995752721271257, −2.42528560481600675660778479463, −2.13714247094967952376546626788, −2.06496465138367239164539094911, −1.95809157833503062367994309952, −1.79685773660640765526976776014, −1.70901048235214580773002107770, −1.56136625699401645190843701220, −1.50566004890197520431679732129, −1.45372833836443366349240705649, −0.860763009186747042563098181198, −0.812710785212460424056256829853, −0.72437725909974954676258127974, −0.57518495177276648831978686307, −0.54757119490590729770638961123, −0.44857458715508367212110715427, −0.31889935448663563386899566756, 0.31889935448663563386899566756, 0.44857458715508367212110715427, 0.54757119490590729770638961123, 0.57518495177276648831978686307, 0.72437725909974954676258127974, 0.812710785212460424056256829853, 0.860763009186747042563098181198, 1.45372833836443366349240705649, 1.50566004890197520431679732129, 1.56136625699401645190843701220, 1.70901048235214580773002107770, 1.79685773660640765526976776014, 1.95809157833503062367994309952, 2.06496465138367239164539094911, 2.13714247094967952376546626788, 2.42528560481600675660778479463, 2.50499634956667995752721271257, 2.65850305039670375424294482375, 2.74801252452625671302720340078, 2.76763016875216247281196840499, 2.95622444013472081082784267488, 2.96003783906370937210071995691, 2.96557872015502756373934619578, 3.17490984374461120607307955162, 3.60795758996711955335859338440
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.