Dirichlet series
| L(s) = 1 | − 8·3-s − 16·4-s + 30·9-s + 128·12-s + 144·16-s + 48·19-s − 40·25-s − 48·27-s − 480·36-s + 80·37-s − 336·43-s − 1.15e3·48-s + 56·49-s − 384·57-s + 112·61-s − 960·64-s + 240·67-s + 48·73-s + 320·75-s − 768·76-s + 8·79-s − 93·81-s − 192·97-s + 640·100-s + 464·103-s + 768·108-s − 312·109-s + ⋯ |
| L(s) = 1 | − 8/3·3-s − 4·4-s + 10/3·9-s + 32/3·12-s + 9·16-s + 2.52·19-s − 8/5·25-s − 1.77·27-s − 13.3·36-s + 2.16·37-s − 7.81·43-s − 24·48-s + 8/7·49-s − 6.73·57-s + 1.83·61-s − 15·64-s + 3.58·67-s + 0.657·73-s + 4.26·75-s − 10.1·76-s + 8/79·79-s − 1.14·81-s − 1.97·97-s + 32/5·100-s + 4.50·103-s + 64/9·108-s − 2.86·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.32089\times 10^{12}\) |
| Root analytic conductor: | \(2.39208\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.7793398542\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7793398542\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T^{2} )^{8} \) |
| 3 | \( 1 + 8 T + 34 T^{2} + 80 T^{3} + 97 T^{4} + 80 T^{5} + 166 p T^{6} + 1096 p T^{7} + 1316 p^{2} T^{8} + 1096 p^{3} T^{9} + 166 p^{5} T^{10} + 80 p^{6} T^{11} + 97 p^{8} T^{12} + 80 p^{10} T^{13} + 34 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 5 | \( ( 1 + p T^{2} )^{8} \) | |
| 7 | \( ( 1 - p T^{2} )^{8} \) | |
| good | 11 | \( 1 - 588 T^{2} + 192110 T^{4} - 45729776 T^{6} + 8909879425 T^{8} - 12537686104 p^{2} T^{10} + 233462822472086 T^{12} - 32740893640977644 T^{14} + 4166709531307795396 T^{16} - 32740893640977644 p^{4} T^{18} + 233462822472086 p^{8} T^{20} - 12537686104 p^{14} T^{22} + 8909879425 p^{16} T^{24} - 45729776 p^{20} T^{26} + 192110 p^{24} T^{28} - 588 p^{28} T^{30} + p^{32} T^{32} \) |
| 13 | \( ( 1 + 38 p T^{2} - 456 T^{3} + 146293 T^{4} - 55296 T^{5} + 35813146 T^{6} - 25079880 T^{7} + 6821269256 T^{8} - 25079880 p^{2} T^{9} + 35813146 p^{4} T^{10} - 55296 p^{6} T^{11} + 146293 p^{8} T^{12} - 456 p^{10} T^{13} + 38 p^{13} T^{14} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 - 2012 T^{2} + 2016998 T^{4} - 1352297560 T^{6} + 695200323953 T^{8} - 299701490294440 T^{10} + 114417537477919350 T^{12} - 39314188029348430308 T^{14} + \)\(12\!\cdots\!52\)\( T^{16} - 39314188029348430308 p^{4} T^{18} + 114417537477919350 p^{8} T^{20} - 299701490294440 p^{12} T^{22} + 695200323953 p^{16} T^{24} - 1352297560 p^{20} T^{26} + 2016998 p^{24} T^{28} - 2012 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 - 24 T + 1612 T^{2} - 39608 T^{3} + 1389160 T^{4} - 32391560 T^{5} + 808980068 T^{6} - 17015640488 T^{7} + 341271444494 T^{8} - 17015640488 p^{2} T^{9} + 808980068 p^{4} T^{10} - 32391560 p^{6} T^{11} + 1389160 p^{8} T^{12} - 39608 p^{10} T^{13} + 1612 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 3856 T^{2} + 7749432 T^{4} - 10915385904 T^{6} + 12001409108892 T^{8} - 10839581553458448 T^{10} + 8275120566094665352 T^{12} - \)\(54\!\cdots\!60\)\( T^{14} + \)\(30\!\cdots\!06\)\( T^{16} - \)\(54\!\cdots\!60\)\( p^{4} T^{18} + 8275120566094665352 p^{8} T^{20} - 10839581553458448 p^{12} T^{22} + 12001409108892 p^{16} T^{24} - 10915385904 p^{20} T^{26} + 7749432 p^{24} T^{28} - 3856 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( 1 - 3972 T^{2} + 8120294 T^{4} - 13034144872 T^{6} + 18580208769649 T^{8} - 22683669812342744 T^{10} + 24087258397916480502 T^{12} - \)\(23\!\cdots\!44\)\( T^{14} + \)\(20\!\cdots\!28\)\( T^{16} - \)\(23\!\cdots\!44\)\( p^{4} T^{18} + 24087258397916480502 p^{8} T^{20} - 22683669812342744 p^{12} T^{22} + 18580208769649 p^{16} T^{24} - 13034144872 p^{20} T^{26} + 8120294 p^{24} T^{28} - 3972 p^{28} T^{30} + p^{32} T^{32} \) | |
| 31 | \( ( 1 + 3684 T^{2} - 19472 T^{3} + 7802120 T^{4} - 58737456 T^{5} + 11372951532 T^{6} - 3015382720 p T^{7} + 12493787155086 T^{8} - 3015382720 p^{3} T^{9} + 11372951532 p^{4} T^{10} - 58737456 p^{6} T^{11} + 7802120 p^{8} T^{12} - 19472 p^{10} T^{13} + 3684 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 40 T + 6968 T^{2} - 291512 T^{3} + 25042268 T^{4} - 987943976 T^{5} + 58242343432 T^{6} - 2052793579384 T^{7} + 94406736665478 T^{8} - 2052793579384 p^{2} T^{9} + 58242343432 p^{4} T^{10} - 987943976 p^{6} T^{11} + 25042268 p^{8} T^{12} - 291512 p^{10} T^{13} + 6968 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( 1 - 10832 T^{2} + 65454200 T^{4} - 277210214128 T^{6} + 920889789641756 T^{8} - 2528312319464856400 T^{10} + \)\(59\!\cdots\!28\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{14} + \)\(21\!\cdots\!66\)\( T^{16} - \)\(12\!\cdots\!00\)\( p^{4} T^{18} + \)\(59\!\cdots\!28\)\( p^{8} T^{20} - 2528312319464856400 p^{12} T^{22} + 920889789641756 p^{16} T^{24} - 277210214128 p^{20} T^{26} + 65454200 p^{24} T^{28} - 10832 p^{28} T^{30} + p^{32} T^{32} \) | |
| 43 | \( ( 1 + 168 T + 21576 T^{2} + 1867768 T^{3} + 137965244 T^{4} + 8243885160 T^{5} + 447366405624 T^{6} + 21255986843000 T^{7} + 961142451546054 T^{8} + 21255986843000 p^{2} T^{9} + 447366405624 p^{4} T^{10} + 8243885160 p^{6} T^{11} + 137965244 p^{8} T^{12} + 1867768 p^{10} T^{13} + 21576 p^{12} T^{14} + 168 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 18860 T^{2} + 165198086 T^{4} - 880812409480 T^{6} + 3099424527294161 T^{8} - 7043887723434328360 T^{10} + \)\(79\!\cdots\!02\)\( T^{12} + \)\(68\!\cdots\!80\)\( T^{14} - \)\(40\!\cdots\!04\)\( T^{16} + \)\(68\!\cdots\!80\)\( p^{4} T^{18} + \)\(79\!\cdots\!02\)\( p^{8} T^{20} - 7043887723434328360 p^{12} T^{22} + 3099424527294161 p^{16} T^{24} - 880812409480 p^{20} T^{26} + 165198086 p^{24} T^{28} - 18860 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 - 15704 T^{2} + 138921152 T^{4} - 871766999944 T^{6} + 4342207935569660 T^{8} - 18113936181752364376 T^{10} + \)\(65\!\cdots\!52\)\( T^{12} - \)\(21\!\cdots\!64\)\( T^{14} + \)\(62\!\cdots\!50\)\( T^{16} - \)\(21\!\cdots\!64\)\( p^{4} T^{18} + \)\(65\!\cdots\!52\)\( p^{8} T^{20} - 18113936181752364376 p^{12} T^{22} + 4342207935569660 p^{16} T^{24} - 871766999944 p^{20} T^{26} + 138921152 p^{24} T^{28} - 15704 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( 1 - 26768 T^{2} + 368873336 T^{4} - 59349485840 p T^{6} + 25676127920457884 T^{8} - \)\(15\!\cdots\!96\)\( T^{10} + \)\(77\!\cdots\!60\)\( T^{12} - \)\(33\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!58\)\( T^{16} - \)\(33\!\cdots\!08\)\( p^{4} T^{18} + \)\(77\!\cdots\!60\)\( p^{8} T^{20} - \)\(15\!\cdots\!96\)\( p^{12} T^{22} + 25676127920457884 p^{16} T^{24} - 59349485840 p^{21} T^{26} + 368873336 p^{24} T^{28} - 26768 p^{28} T^{30} + p^{32} T^{32} \) | |
| 61 | \( ( 1 - 56 T + 11072 T^{2} - 269288 T^{3} + 50054524 T^{4} - 711106488 T^{5} + 4064218048 p T^{6} - 7112512171688 T^{7} + 1188015110923334 T^{8} - 7112512171688 p^{2} T^{9} + 4064218048 p^{5} T^{10} - 711106488 p^{6} T^{11} + 50054524 p^{8} T^{12} - 269288 p^{10} T^{13} + 11072 p^{12} T^{14} - 56 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 120 T + 21032 T^{2} - 1363784 T^{3} + 159985756 T^{4} - 7095679000 T^{5} + 808761061016 T^{6} - 25648223522216 T^{7} + 3474267779595526 T^{8} - 25648223522216 p^{2} T^{9} + 808761061016 p^{4} T^{10} - 7095679000 p^{6} T^{11} + 159985756 p^{8} T^{12} - 1363784 p^{10} T^{13} + 21032 p^{12} T^{14} - 120 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( 1 - 58328 T^{2} + 1630496768 T^{4} - 29214173879944 T^{6} + 378588472445657852 T^{8} - \)\(37\!\cdots\!40\)\( T^{10} + \)\(30\!\cdots\!12\)\( T^{12} - \)\(20\!\cdots\!48\)\( T^{14} + \)\(11\!\cdots\!90\)\( T^{16} - \)\(20\!\cdots\!48\)\( p^{4} T^{18} + \)\(30\!\cdots\!12\)\( p^{8} T^{20} - \)\(37\!\cdots\!40\)\( p^{12} T^{22} + 378588472445657852 p^{16} T^{24} - 29214173879944 p^{20} T^{26} + 1630496768 p^{24} T^{28} - 58328 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 - 24 T + 15156 T^{2} + 137320 T^{3} + 133090184 T^{4} + 1190310168 T^{5} + 1034805205980 T^{6} + 7023266129048 T^{7} + 5908298409616782 T^{8} + 7023266129048 p^{2} T^{9} + 1034805205980 p^{4} T^{10} + 1190310168 p^{6} T^{11} + 133090184 p^{8} T^{12} + 137320 p^{10} T^{13} + 15156 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 4 T + 21374 T^{2} - 1216 T^{3} + 230456449 T^{4} + 2685965400 T^{5} + 1772108711446 T^{6} + 44455008585260 T^{7} + 11672772046344164 T^{8} + 44455008585260 p^{2} T^{9} + 1772108711446 p^{4} T^{10} + 2685965400 p^{6} T^{11} + 230456449 p^{8} T^{12} - 1216 p^{10} T^{13} + 21374 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( 1 - 26752 T^{2} + 559157880 T^{4} - 7483229844864 T^{6} + 85634770716597660 T^{8} - \)\(74\!\cdots\!16\)\( T^{10} + \)\(60\!\cdots\!76\)\( T^{12} - \)\(41\!\cdots\!76\)\( T^{14} + \)\(29\!\cdots\!06\)\( T^{16} - \)\(41\!\cdots\!76\)\( p^{4} T^{18} + \)\(60\!\cdots\!76\)\( p^{8} T^{20} - \)\(74\!\cdots\!16\)\( p^{12} T^{22} + 85634770716597660 p^{16} T^{24} - 7483229844864 p^{20} T^{26} + 559157880 p^{24} T^{28} - 26752 p^{28} T^{30} + p^{32} T^{32} \) | |
| 89 | \( 1 - 71136 T^{2} + 2535583160 T^{4} - 60654761408672 T^{6} + 1094469962087760028 T^{8} - \)\(15\!\cdots\!12\)\( T^{10} + \)\(18\!\cdots\!28\)\( T^{12} - \)\(19\!\cdots\!80\)\( T^{14} + \)\(16\!\cdots\!02\)\( T^{16} - \)\(19\!\cdots\!80\)\( p^{4} T^{18} + \)\(18\!\cdots\!28\)\( p^{8} T^{20} - \)\(15\!\cdots\!12\)\( p^{12} T^{22} + 1094469962087760028 p^{16} T^{24} - 60654761408672 p^{20} T^{26} + 2535583160 p^{24} T^{28} - 71136 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( ( 1 + 96 T + 62878 T^{2} + 5357144 T^{3} + 1785824677 T^{4} + 133941251648 T^{5} + 30441979716138 T^{6} + 1968996759462584 T^{7} + 345736614004039176 T^{8} + 1968996759462584 p^{2} T^{9} + 30441979716138 p^{4} T^{10} + 133941251648 p^{6} T^{11} + 1785824677 p^{8} T^{12} + 5357144 p^{10} T^{13} + 62878 p^{12} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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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
−3.45782537927774142727587528075, −3.20364381914309351141482512921, −3.16904763739131390221598821674, −3.08467206455967005115956684836, −2.99394207117712309538951045503, −2.95718588286098437254588925749, −2.66543338844783644152305214348, −2.61737021093543163568465576316, −2.49352238858051600073136508712, −2.40748967736049503428288120503, −2.33313609630127905847552160923, −1.88534670401868889933467955498, −1.82424699884256681637419674414, −1.77982705880394792913036832233, −1.68427044924047643697294176097, −1.61560320777198558239218585429, −1.46759390101367870892409673974, −1.22693387444272117522431927911, −0.988316204558238178994536696953, −0.821742219874183493469639748733, −0.67554090338942628777988336272, −0.60716155965645057387061792344, −0.54874286670401732033505386752, −0.31996375276973764545148672082, −0.23025089808649407265884754722, 0.23025089808649407265884754722, 0.31996375276973764545148672082, 0.54874286670401732033505386752, 0.60716155965645057387061792344, 0.67554090338942628777988336272, 0.821742219874183493469639748733, 0.988316204558238178994536696953, 1.22693387444272117522431927911, 1.46759390101367870892409673974, 1.61560320777198558239218585429, 1.68427044924047643697294176097, 1.77982705880394792913036832233, 1.82424699884256681637419674414, 1.88534670401868889933467955498, 2.33313609630127905847552160923, 2.40748967736049503428288120503, 2.49352238858051600073136508712, 2.61737021093543163568465576316, 2.66543338844783644152305214348, 2.95718588286098437254588925749, 2.99394207117712309538951045503, 3.08467206455967005115956684836, 3.16904763739131390221598821674, 3.20364381914309351141482512921, 3.45782537927774142727587528075
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.