Dirichlet series
| L(s) = 1 | + 6·5-s − 10·9-s − 11-s − 2·23-s − 5·25-s − 12·31-s + 2·37-s − 60·45-s − 12·47-s − 2·49-s + 8·53-s − 6·55-s − 30·59-s − 18·67-s − 76·71-s + 51·81-s − 12·89-s + 10·99-s − 54·103-s + 60·113-s − 12·115-s + 8·121-s − 102·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 3.33·9-s − 0.301·11-s − 0.417·23-s − 25-s − 2.15·31-s + 0.328·37-s − 8.94·45-s − 1.75·47-s − 2/7·49-s + 1.09·53-s − 0.809·55-s − 3.90·59-s − 2.19·67-s − 9.01·71-s + 17/3·81-s − 1.27·89-s + 1.00·99-s − 5.32·103-s + 5.64·113-s − 1.11·115-s + 8/11·121-s − 9.12·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 7^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.79161\times 10^{6}\) |
| Root analytic conductor: | \(1.56824\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 7^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.01211076416\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01211076416\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2 T^{2} - 17 T^{4} - p^{3} T^{6} + 10 p^{2} T^{8} - p^{5} T^{10} - 17 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) | |
| 11 | \( 1 + T - 7 T^{2} + 30 T^{3} + 101 T^{4} - 157 T^{5} + 2386 T^{6} + 411 p T^{7} - 20378 T^{8} + 411 p^{2} T^{9} + 2386 p^{2} T^{10} - 157 p^{3} T^{11} + 101 p^{4} T^{12} + 30 p^{5} T^{13} - 7 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( ( 1 + 5 T^{2} + 4 p T^{4} - 2 p T^{5} - 22 T^{6} - 13 p T^{7} - 128 T^{8} - 13 p^{2} T^{9} - 22 p^{2} T^{10} - 2 p^{4} T^{11} + 4 p^{5} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 5 | \( ( 1 - 3 T + 16 T^{2} - 39 T^{3} + 133 T^{4} - 72 p T^{5} + 853 T^{6} - 459 p T^{7} + 4336 T^{8} - 459 p^{2} T^{9} + 853 p^{2} T^{10} - 72 p^{4} T^{11} + 133 p^{4} T^{12} - 39 p^{5} T^{13} + 16 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 9 T^{2} + 556 T^{4} + 3678 T^{6} + 130224 T^{8} + 3678 p^{2} T^{10} + 556 p^{4} T^{12} + 9 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 44 T^{2} + 33 p T^{4} - 5638 T^{6} + 162806 T^{8} + 190056 T^{10} - 82845710 T^{12} + 1398140209 T^{14} - 17592920802 T^{16} + 1398140209 p^{2} T^{18} - 82845710 p^{4} T^{20} + 190056 p^{6} T^{22} + 162806 p^{8} T^{24} - 5638 p^{10} T^{26} + 33 p^{13} T^{28} - 44 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 60 T^{2} + 1313 T^{4} - 20454 T^{6} + 479962 T^{8} - 4906944 T^{10} - 112778386 T^{12} + 2868895941 T^{14} - 34301480690 T^{16} + 2868895941 p^{2} T^{18} - 112778386 p^{4} T^{20} - 4906944 p^{6} T^{22} + 479962 p^{8} T^{24} - 20454 p^{10} T^{26} + 1313 p^{12} T^{28} - 60 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + T - 58 T^{2} - 39 T^{3} + 1565 T^{4} + 284 T^{5} - 46457 T^{6} + 1875 T^{7} + 1333510 T^{8} + 1875 p T^{9} - 46457 p^{2} T^{10} + 284 p^{3} T^{11} + 1565 p^{4} T^{12} - 39 p^{5} T^{13} - 58 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 37 T^{2} + 1630 T^{4} - 43912 T^{6} + 1578436 T^{8} - 43912 p^{2} T^{10} + 1630 p^{4} T^{12} - 37 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 6 T + 75 T^{2} + 378 T^{3} + 1894 T^{4} + 12768 T^{5} + 88242 T^{6} + 604323 T^{7} + 4446492 T^{8} + 604323 p T^{9} + 88242 p^{2} T^{10} + 12768 p^{3} T^{11} + 1894 p^{4} T^{12} + 378 p^{5} T^{13} + 75 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - T - 3 p T^{2} + 22 T^{3} + 6887 T^{4} + 1197 T^{5} - 317414 T^{6} - 23215 T^{7} + 12136416 T^{8} - 23215 p T^{9} - 317414 p^{2} T^{10} + 1197 p^{3} T^{11} + 6887 p^{4} T^{12} + 22 p^{5} T^{13} - 3 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 233 T^{2} + 26932 T^{4} + 1947830 T^{6} + 95882440 T^{8} + 1947830 p^{2} T^{10} + 26932 p^{4} T^{12} + 233 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 136 T^{2} + 6319 T^{4} - 77821 T^{6} - 1877150 T^{8} - 77821 p^{2} T^{10} + 6319 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 6 T + 127 T^{2} + 690 T^{3} + 7288 T^{4} + 22236 T^{5} + 254800 T^{6} + 8109 T^{7} + 8531044 T^{8} + 8109 p T^{9} + 254800 p^{2} T^{10} + 22236 p^{3} T^{11} + 7288 p^{4} T^{12} + 690 p^{5} T^{13} + 127 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 - 4 T - 73 T^{2} + 1350 T^{3} - 238 T^{4} - 83150 T^{5} + 624292 T^{6} + 2402499 T^{7} - 41405774 T^{8} + 2402499 p T^{9} + 624292 p^{2} T^{10} - 83150 p^{3} T^{11} - 238 p^{4} T^{12} + 1350 p^{5} T^{13} - 73 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( ( 1 + 15 T + 244 T^{2} + 2535 T^{3} + 23995 T^{4} + 151470 T^{5} + 984469 T^{6} + 3863007 T^{7} + 26070886 T^{8} + 3863007 p T^{9} + 984469 p^{2} T^{10} + 151470 p^{3} T^{11} + 23995 p^{4} T^{12} + 2535 p^{5} T^{13} + 244 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 179 T^{2} + 9951 T^{4} - 101332 T^{6} + 106631 p T^{8} - 804914355 T^{10} + 21217989730 T^{12} - 10505506486151 T^{14} + 1215439769319264 T^{16} - 10505506486151 p^{2} T^{18} + 21217989730 p^{4} T^{20} - 804914355 p^{6} T^{22} + 106631 p^{9} T^{24} - 101332 p^{10} T^{26} + 9951 p^{12} T^{28} - 179 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 9 T - 139 T^{2} - 1086 T^{3} + 13837 T^{4} + 66183 T^{5} - 1167670 T^{6} - 1763475 T^{7} + 85075678 T^{8} - 1763475 p T^{9} - 1167670 p^{2} T^{10} + 66183 p^{3} T^{11} + 13837 p^{4} T^{12} - 1086 p^{5} T^{13} - 139 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 19 T + 362 T^{2} + 3934 T^{3} + 41338 T^{4} + 3934 p T^{5} + 362 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 - 143 T^{2} + 156 T^{4} + 1111775 T^{6} - 32820655 T^{8} - 6331617096 T^{10} + 451494276247 T^{12} + 13619221009111 T^{14} - 2920593003737400 T^{16} + 13619221009111 p^{2} T^{18} + 451494276247 p^{4} T^{20} - 6331617096 p^{6} T^{22} - 32820655 p^{8} T^{24} + 1111775 p^{10} T^{26} + 156 p^{12} T^{28} - 143 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 + 298 T^{2} + 39591 T^{4} + 2909420 T^{6} + 134099840 T^{8} + 6404289390 T^{10} + 1066431325270 T^{12} + 184533575326711 T^{14} + 19164734201961918 T^{16} + 184533575326711 p^{2} T^{18} + 1066431325270 p^{4} T^{20} + 6404289390 p^{6} T^{22} + 134099840 p^{8} T^{24} + 2909420 p^{10} T^{26} + 39591 p^{12} T^{28} + 298 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 401 T^{2} + 71914 T^{4} + 8121998 T^{6} + 721115524 T^{8} + 8121998 p^{2} T^{10} + 71914 p^{4} T^{12} + 401 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 6 T + 185 T^{2} + 1038 T^{3} + 17980 T^{4} + 203364 T^{5} + 1686986 T^{6} + 28199199 T^{7} + 139092058 T^{8} + 28199199 p T^{9} + 1686986 p^{2} T^{10} + 203364 p^{3} T^{11} + 17980 p^{4} T^{12} + 1038 p^{5} T^{13} + 185 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 585 T^{2} + 164188 T^{4} - 28521612 T^{6} + 3334554816 T^{8} - 28521612 p^{2} T^{10} + 164188 p^{4} T^{12} - 585 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−3.29881860228607185953506089972, −3.09317328039170647334418186058, −3.04860437335556463841504433914, −2.99126796315310902024173392978, −2.98664127162760410073659544401, −2.92194928004495349444214649879, −2.74773274474901339738615140967, −2.71417261809745430954273481386, −2.56038939153914606716849843383, −2.49227280222550178023509860276, −2.24385431739592077489791990723, −2.19296155572698749659032708789, −2.18095750505750988718642687447, −2.05462708163070462321469979218, −1.85239528015002105025194216150, −1.85064495937244902856569411096, −1.65910498345656642534610477760, −1.58824942248649906678314895029, −1.58687360721663388984531872858, −1.50064323044594585941767350321, −1.27922033866894008437360158037, −1.05897606197713632249664704893, −0.69884957718535691083527582901, −0.14573466219951513878369454806, −0.04703815008283968495498614210, 0.04703815008283968495498614210, 0.14573466219951513878369454806, 0.69884957718535691083527582901, 1.05897606197713632249664704893, 1.27922033866894008437360158037, 1.50064323044594585941767350321, 1.58687360721663388984531872858, 1.58824942248649906678314895029, 1.65910498345656642534610477760, 1.85064495937244902856569411096, 1.85239528015002105025194216150, 2.05462708163070462321469979218, 2.18095750505750988718642687447, 2.19296155572698749659032708789, 2.24385431739592077489791990723, 2.49227280222550178023509860276, 2.56038939153914606716849843383, 2.71417261809745430954273481386, 2.74773274474901339738615140967, 2.92194928004495349444214649879, 2.98664127162760410073659544401, 2.99126796315310902024173392978, 3.04860437335556463841504433914, 3.09317328039170647334418186058, 3.29881860228607185953506089972
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.