Dirichlet series
| L(s) = 1 | + 13·4-s + 3·5-s + 20·9-s − 16·11-s + 77·16-s − 16·19-s + 39·20-s + 25-s + 2·29-s − 32·31-s + 260·36-s + 6·41-s − 208·44-s + 60·45-s + 75·49-s − 48·55-s + 24·59-s − 42·61-s + 273·64-s − 46·71-s − 208·76-s + 74·79-s + 231·80-s + 180·81-s + 14·89-s − 48·95-s − 320·99-s + ⋯ |
| L(s) = 1 | + 13/2·4-s + 1.34·5-s + 20/3·9-s − 4.82·11-s + 77/4·16-s − 3.67·19-s + 8.72·20-s + 1/5·25-s + 0.371·29-s − 5.74·31-s + 43.3·36-s + 0.937·41-s − 31.3·44-s + 8.94·45-s + 75/7·49-s − 6.47·55-s + 3.12·59-s − 5.37·61-s + 34.1·64-s − 5.45·71-s − 23.8·76-s + 8.32·79-s + 25.8·80-s + 20·81-s + 1.48·89-s − 4.92·95-s − 32.1·99-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{16} \cdot 11^{16} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.52451\times 10^{14}\) |
| Root analytic conductor: | \(2.88866\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 5^{16} \cdot 11^{16} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(191.1539614\) |
| \(L(\frac12)\) | \(\approx\) | \(191.1539614\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - 3 T + 8 T^{2} - p T^{3} + 4 T^{4} + 29 T^{5} + 152 T^{6} - 541 T^{7} + 1942 T^{8} - 541 p T^{9} + 152 p^{2} T^{10} + 29 p^{3} T^{11} + 4 p^{4} T^{12} - p^{6} T^{13} + 8 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( ( 1 + T )^{16} \) | |
| 19 | \( ( 1 + T )^{16} \) | |
| good | 2 | \( 1 - 13 T^{2} + 23 p^{2} T^{4} - 117 p^{2} T^{6} + 1887 T^{8} - 3155 p T^{10} + 4489 p^{2} T^{12} - 44149 T^{14} + 11815 p^{3} T^{16} - 44149 p^{2} T^{18} + 4489 p^{6} T^{20} - 3155 p^{7} T^{22} + 1887 p^{8} T^{24} - 117 p^{12} T^{26} + 23 p^{14} T^{28} - 13 p^{14} T^{30} + p^{16} T^{32} \) |
| 3 | \( 1 - 20 T^{2} + 220 T^{4} - 1715 T^{6} + 10453 T^{8} - 52535 T^{10} + 224215 T^{12} - 275470 p T^{14} + 2653246 T^{16} - 275470 p^{3} T^{18} + 224215 p^{4} T^{20} - 52535 p^{6} T^{22} + 10453 p^{8} T^{24} - 1715 p^{10} T^{26} + 220 p^{12} T^{28} - 20 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( 1 - 75 T^{2} + 2791 T^{4} - 68297 T^{6} + 1228343 T^{8} - 17200826 T^{10} + 193881783 T^{12} - 1793786450 T^{14} + 13764879668 T^{16} - 1793786450 p^{2} T^{18} + 193881783 p^{4} T^{20} - 17200826 p^{6} T^{22} + 1228343 p^{8} T^{24} - 68297 p^{10} T^{26} + 2791 p^{12} T^{28} - 75 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 - 127 T^{2} + 7825 T^{4} - 311914 T^{6} + 9066140 T^{8} - 205895579 T^{10} + 3834814053 T^{12} - 60793187292 T^{14} + 841674980890 T^{16} - 60793187292 p^{2} T^{18} + 3834814053 p^{4} T^{20} - 205895579 p^{6} T^{22} + 9066140 p^{8} T^{24} - 311914 p^{10} T^{26} + 7825 p^{12} T^{28} - 127 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 159 T^{2} + 12629 T^{4} - 664606 T^{6} + 25984300 T^{8} - 803375391 T^{10} + 20421930993 T^{12} - 25746037892 p T^{14} + 8024997757786 T^{16} - 25746037892 p^{3} T^{18} + 20421930993 p^{4} T^{20} - 803375391 p^{6} T^{22} + 25984300 p^{8} T^{24} - 664606 p^{10} T^{26} + 12629 p^{12} T^{28} - 159 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 187 T^{2} + 16318 T^{4} - 893208 T^{6} + 35352343 T^{8} - 1132402989 T^{10} + 1405119437 p T^{12} - 857540530052 T^{14} + 20807981893670 T^{16} - 857540530052 p^{2} T^{18} + 1405119437 p^{5} T^{20} - 1132402989 p^{6} T^{22} + 35352343 p^{8} T^{24} - 893208 p^{10} T^{26} + 16318 p^{12} T^{28} - 187 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - T + 173 T^{2} - 52 T^{3} + 13852 T^{4} + 2183 T^{5} + 688395 T^{6} + 239734 T^{7} + 23651094 T^{8} + 239734 p T^{9} + 688395 p^{2} T^{10} + 2183 p^{3} T^{11} + 13852 p^{4} T^{12} - 52 p^{5} T^{13} + 173 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 16 T + 254 T^{2} + 2500 T^{3} + 24395 T^{4} + 181079 T^{5} + 1345289 T^{6} + 8144987 T^{7} + 49804410 T^{8} + 8144987 p T^{9} + 1345289 p^{2} T^{10} + 181079 p^{3} T^{11} + 24395 p^{4} T^{12} + 2500 p^{5} T^{13} + 254 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 388 T^{2} + 74501 T^{4} - 9409292 T^{6} + 875599833 T^{8} - 1721308464 p T^{10} + 3746350584006 T^{12} - 181798143394000 T^{14} + 7354842347001414 T^{16} - 181798143394000 p^{2} T^{18} + 3746350584006 p^{4} T^{20} - 1721308464 p^{7} T^{22} + 875599833 p^{8} T^{24} - 9409292 p^{10} T^{26} + 74501 p^{12} T^{28} - 388 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 3 T + 260 T^{2} - 704 T^{3} + 31731 T^{4} - 76143 T^{5} + 2365080 T^{6} - 4882558 T^{7} + 117441184 T^{8} - 4882558 p T^{9} + 2365080 p^{2} T^{10} - 76143 p^{3} T^{11} + 31731 p^{4} T^{12} - 704 p^{5} T^{13} + 260 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 373 T^{2} + 69047 T^{4} - 8513085 T^{6} + 790370777 T^{8} - 59029616338 T^{10} + 3685526806951 T^{12} - 196695248354404 T^{14} + 9078456827272144 T^{16} - 196695248354404 p^{2} T^{18} + 3685526806951 p^{4} T^{20} - 59029616338 p^{6} T^{22} + 790370777 p^{8} T^{24} - 8513085 p^{10} T^{26} + 69047 p^{12} T^{28} - 373 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 - 378 T^{2} + 67480 T^{4} - 7503137 T^{6} + 575685206 T^{8} - 31742514468 T^{10} + 1271141328010 T^{12} - 824982864255 p T^{14} + 1324031301162926 T^{16} - 824982864255 p^{3} T^{18} + 1271141328010 p^{4} T^{20} - 31742514468 p^{6} T^{22} + 575685206 p^{8} T^{24} - 7503137 p^{10} T^{26} + 67480 p^{12} T^{28} - 378 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 479 T^{2} + 115345 T^{4} - 18634182 T^{6} + 2267490560 T^{8} - 220483232255 T^{10} + 17703325159429 T^{12} - 22564719810796 p T^{14} + 24439711186722 p^{2} T^{16} - 22564719810796 p^{3} T^{18} + 17703325159429 p^{4} T^{20} - 220483232255 p^{6} T^{22} + 2267490560 p^{8} T^{24} - 18634182 p^{10} T^{26} + 115345 p^{12} T^{28} - 479 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 12 T + 222 T^{2} - 2458 T^{3} + 29923 T^{4} - 288525 T^{5} + 2646021 T^{6} - 22878075 T^{7} + 180833418 T^{8} - 22878075 p T^{9} + 2646021 p^{2} T^{10} - 288525 p^{3} T^{11} + 29923 p^{4} T^{12} - 2458 p^{5} T^{13} + 222 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 21 T + 491 T^{2} + 5754 T^{3} + 72440 T^{4} + 525253 T^{5} + 4647577 T^{6} + 22460636 T^{7} + 223379062 T^{8} + 22460636 p T^{9} + 4647577 p^{2} T^{10} + 525253 p^{3} T^{11} + 72440 p^{4} T^{12} + 5754 p^{5} T^{13} + 491 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 593 T^{2} + 178505 T^{4} - 36210316 T^{6} + 5545820776 T^{8} - 679682844487 T^{10} + 68858755146492 T^{12} - 5874388897293084 T^{14} + 426153464891454484 T^{16} - 5874388897293084 p^{2} T^{18} + 68858755146492 p^{4} T^{20} - 679682844487 p^{6} T^{22} + 5545820776 p^{8} T^{24} - 36210316 p^{10} T^{26} + 178505 p^{12} T^{28} - 593 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 23 T + 540 T^{2} + 8634 T^{3} + 127767 T^{4} + 1551201 T^{5} + 17433009 T^{6} + 169640126 T^{7} + 1524083270 T^{8} + 169640126 p T^{9} + 17433009 p^{2} T^{10} + 1551201 p^{3} T^{11} + 127767 p^{4} T^{12} + 8634 p^{5} T^{13} + 540 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 - 524 T^{2} + 150729 T^{4} - 30560292 T^{6} + 4814238248 T^{8} - 618415562343 T^{10} + 66568684535969 T^{12} - 6101248902416617 T^{14} + 480064827878639034 T^{16} - 6101248902416617 p^{2} T^{18} + 66568684535969 p^{4} T^{20} - 618415562343 p^{6} T^{22} + 4814238248 p^{8} T^{24} - 30560292 p^{10} T^{26} + 150729 p^{12} T^{28} - 524 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 37 T + 960 T^{2} - 229 p T^{3} + 281561 T^{4} - 3686856 T^{5} + 42550550 T^{6} - 436806918 T^{7} + 4077420768 T^{8} - 436806918 p T^{9} + 42550550 p^{2} T^{10} - 3686856 p^{3} T^{11} + 281561 p^{4} T^{12} - 229 p^{6} T^{13} + 960 p^{6} T^{14} - 37 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 663 T^{2} + 231697 T^{4} - 55676590 T^{6} + 10207807056 T^{8} - 1507976909199 T^{10} + 185375309050265 T^{12} - 19336808869817660 T^{14} + 1730030032689959594 T^{16} - 19336808869817660 p^{2} T^{18} + 185375309050265 p^{4} T^{20} - 1507976909199 p^{6} T^{22} + 10207807056 p^{8} T^{24} - 55676590 p^{10} T^{26} + 231697 p^{12} T^{28} - 663 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 7 T + 549 T^{2} - 3742 T^{3} + 143826 T^{4} - 892465 T^{5} + 23215882 T^{6} - 124792550 T^{7} + 2502496900 T^{8} - 124792550 p T^{9} + 23215882 p^{2} T^{10} - 892465 p^{3} T^{11} + 143826 p^{4} T^{12} - 3742 p^{5} T^{13} + 549 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 555 T^{2} + 154287 T^{4} - 28420296 T^{6} + 4029887320 T^{8} - 500111019121 T^{10} + 59650934547046 T^{12} - 6831004755368268 T^{14} + 707715099596477700 T^{16} - 6831004755368268 p^{2} T^{18} + 59650934547046 p^{4} T^{20} - 500111019121 p^{6} T^{22} + 4029887320 p^{8} T^{24} - 28420296 p^{10} T^{26} + 154287 p^{12} T^{28} - 555 p^{14} T^{30} + p^{16} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.47918037507947566295807095075, −2.43074445902524973203137801936, −2.39915871121573909540101295442, −2.39132440757289506601135319625, −2.19347485082039170853312719423, −2.01949119028132033956398362080, −1.97204766350480311551684938470, −1.93382256473730150170184533965, −1.91641822374129751858341839473, −1.85625075504709973974900087450, −1.83366327790762968709409135623, −1.82932327020950838594195394369, −1.79983933300663721415201622571, −1.78936317226262983416203547024, −1.75178711593464028137921263172, −1.45724806949068192598537227094, −1.42312398857202301979128059956, −1.00362764946805033809207953155, −0.971731814864739029042633358705, −0.968633255006608384086507968799, −0.861274610707743954458580422574, −0.68585100339320585390567118836, −0.67969249760368036503610383658, −0.39881097637704929093112358341, −0.17238632152101059935510347807, 0.17238632152101059935510347807, 0.39881097637704929093112358341, 0.67969249760368036503610383658, 0.68585100339320585390567118836, 0.861274610707743954458580422574, 0.968633255006608384086507968799, 0.971731814864739029042633358705, 1.00362764946805033809207953155, 1.42312398857202301979128059956, 1.45724806949068192598537227094, 1.75178711593464028137921263172, 1.78936317226262983416203547024, 1.79983933300663721415201622571, 1.82932327020950838594195394369, 1.83366327790762968709409135623, 1.85625075504709973974900087450, 1.91641822374129751858341839473, 1.93382256473730150170184533965, 1.97204766350480311551684938470, 2.01949119028132033956398362080, 2.19347485082039170853312719423, 2.39132440757289506601135319625, 2.39915871121573909540101295442, 2.43074445902524973203137801936, 2.47918037507947566295807095075
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.