Dirichlet series
| L(s) = 1 | + 8·2-s + 24·4-s − 2·5-s + 6·7-s − 16·10-s + 32·11-s − 4·13-s + 48·14-s − 240·16-s − 58·17-s + 70·19-s − 48·20-s + 256·22-s + 86·23-s + 174·25-s − 32·26-s + 144·28-s − 212·29-s − 64·31-s − 768·32-s − 464·34-s − 12·35-s − 146·37-s + 560·38-s − 780·41-s + 880·43-s + 768·44-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3·4-s − 0.178·5-s + 0.323·7-s − 0.505·10-s + 0.877·11-s − 0.0853·13-s + 0.916·14-s − 3.75·16-s − 0.827·17-s + 0.845·19-s − 0.536·20-s + 2.48·22-s + 0.779·23-s + 1.39·25-s − 0.241·26-s + 0.971·28-s − 1.35·29-s − 0.370·31-s − 4.24·32-s − 2.34·34-s − 0.0579·35-s − 0.648·37-s + 2.39·38-s − 2.97·41-s + 3.12·43-s + 2.63·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{24} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.12157\times 10^{10}\) |
| Root analytic conductor: | \(4.72257\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.2063532042\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2063532042\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 6 T - p T^{2} - 354 p T^{3} - 1140 p^{2} T^{4} - 354 p^{4} T^{5} - p^{7} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 5 | \( 1 + 2 T - 34 p T^{2} - 912 T^{3} - 6922 T^{4} + 44618 T^{5} - 576312 T^{6} + 915938 p T^{7} + 540349219 T^{8} + 915938 p^{4} T^{9} - 576312 p^{6} T^{10} + 44618 p^{9} T^{11} - 6922 p^{12} T^{12} - 912 p^{15} T^{13} - 34 p^{19} T^{14} + 2 p^{21} T^{15} + p^{24} T^{16} \) |
| 11 | \( 1 - 32 T - 3524 T^{2} + 77568 T^{3} + 8319434 T^{4} - 92921024 T^{5} - 16193852112 T^{6} + 36132028832 T^{7} + 25840028173315 T^{8} + 36132028832 p^{3} T^{9} - 16193852112 p^{6} T^{10} - 92921024 p^{9} T^{11} + 8319434 p^{12} T^{12} + 77568 p^{15} T^{13} - 3524 p^{18} T^{14} - 32 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( ( 1 + 2 T + 3905 T^{2} - 134190 T^{3} + 6991376 T^{4} - 134190 p^{3} T^{5} + 3905 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 17 | \( 1 + 58 T - 8702 T^{2} - 1253208 T^{3} + 17600774 T^{4} + 488456042 p T^{5} + 396908983752 T^{6} - 22332126339646 T^{7} - 2893130337334877 T^{8} - 22332126339646 p^{3} T^{9} + 396908983752 p^{6} T^{10} + 488456042 p^{10} T^{11} + 17600774 p^{12} T^{12} - 1253208 p^{15} T^{13} - 8702 p^{18} T^{14} + 58 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 70 T - 12273 T^{2} + 1205422 T^{3} + 45848873 T^{4} - 308262288 p T^{5} - 343174472426 T^{6} + 6059827390508 T^{7} + 4294435442748282 T^{8} + 6059827390508 p^{3} T^{9} - 343174472426 p^{6} T^{10} - 308262288 p^{10} T^{11} + 45848873 p^{12} T^{12} + 1205422 p^{15} T^{13} - 12273 p^{18} T^{14} - 70 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 - 86 T - 39890 T^{2} + 82632 p T^{3} + 1124840390 T^{4} - 30401287838 T^{5} - 20627892302808 T^{6} + 126588682789742 T^{7} + 295654711270877179 T^{8} + 126588682789742 p^{3} T^{9} - 20627892302808 p^{6} T^{10} - 30401287838 p^{9} T^{11} + 1124840390 p^{12} T^{12} + 82632 p^{16} T^{13} - 39890 p^{18} T^{14} - 86 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( ( 1 + 106 T + 58386 T^{2} + 2107854 T^{3} + 1482347746 T^{4} + 2107854 p^{3} T^{5} + 58386 p^{6} T^{6} + 106 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 31 | \( 1 + 64 T - 75334 T^{2} - 4471888 T^{3} + 3046133861 T^{4} + 143301706760 T^{5} - 80185819225710 T^{6} - 2278462920223944 T^{7} + 1894674291144946204 T^{8} - 2278462920223944 p^{3} T^{9} - 80185819225710 p^{6} T^{10} + 143301706760 p^{9} T^{11} + 3046133861 p^{12} T^{12} - 4471888 p^{15} T^{13} - 75334 p^{18} T^{14} + 64 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 + 146 T - 41037 T^{2} + 24382222 T^{3} + 4523622941 T^{4} - 1003339821252 T^{5} + 404910943728514 T^{6} + 86923322843591240 T^{7} - 13743755171151605190 T^{8} + 86923322843591240 p^{3} T^{9} + 404910943728514 p^{6} T^{10} - 1003339821252 p^{9} T^{11} + 4523622941 p^{12} T^{12} + 24382222 p^{15} T^{13} - 41037 p^{18} T^{14} + 146 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( ( 1 + 390 T + 214610 T^{2} + 60532506 T^{3} + 22214903298 T^{4} + 60532506 p^{3} T^{5} + 214610 p^{6} T^{6} + 390 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 440 T + 352918 T^{2} - 101685224 T^{3} + 43426524751 T^{4} - 101685224 p^{3} T^{5} + 352918 p^{6} T^{6} - 440 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 47 | \( 1 + 306 T - 73958 T^{2} + 59983056 T^{3} + 24769315990 T^{4} - 4026015675846 T^{5} + 2990954943459448 T^{6} + 21838254530911650 p T^{7} - 85885331168320805 p^{2} T^{8} + 21838254530911650 p^{4} T^{9} + 2990954943459448 p^{6} T^{10} - 4026015675846 p^{9} T^{11} + 24769315990 p^{12} T^{12} + 59983056 p^{15} T^{13} - 73958 p^{18} T^{14} + 306 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 + 90 T - 111650 T^{2} - 55596816 T^{3} - 21791681306 T^{4} + 2948884766586 T^{5} + 1995977814148024 T^{6} + 452517461869649058 T^{7} + \)\(17\!\cdots\!27\)\( T^{8} + 452517461869649058 p^{3} T^{9} + 1995977814148024 p^{6} T^{10} + 2948884766586 p^{9} T^{11} - 21791681306 p^{12} T^{12} - 55596816 p^{15} T^{13} - 111650 p^{18} T^{14} + 90 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 148 T - 724172 T^{2} + 46183224 T^{3} + 322992258026 T^{4} - 8773729679068 T^{5} - 98974299460087056 T^{6} + 669594626352126556 T^{7} + \)\(23\!\cdots\!15\)\( T^{8} + 669594626352126556 p^{3} T^{9} - 98974299460087056 p^{6} T^{10} - 8773729679068 p^{9} T^{11} + 322992258026 p^{12} T^{12} + 46183224 p^{15} T^{13} - 724172 p^{18} T^{14} - 148 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 364 T - 627802 T^{2} - 224697736 T^{3} + 3996544229 p T^{4} + 1157855553464 p T^{5} - 64264165920142506 T^{6} - 7095447872603585772 T^{7} + \)\(15\!\cdots\!52\)\( T^{8} - 7095447872603585772 p^{3} T^{9} - 64264165920142506 p^{6} T^{10} + 1157855553464 p^{10} T^{11} + 3996544229 p^{13} T^{12} - 224697736 p^{15} T^{13} - 627802 p^{18} T^{14} + 364 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 954 T - 528629 T^{2} - 389300634 T^{3} + 544584174805 T^{4} + 225418821262920 T^{5} - 201962106024878270 T^{6} - 8847830048260684788 T^{7} + \)\(89\!\cdots\!70\)\( T^{8} - 8847830048260684788 p^{3} T^{9} - 201962106024878270 p^{6} T^{10} + 225418821262920 p^{9} T^{11} + 544584174805 p^{12} T^{12} - 389300634 p^{15} T^{13} - 528629 p^{18} T^{14} + 954 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( ( 1 + 680 T + 1000056 T^{2} + 625468776 T^{3} + 454321868590 T^{4} + 625468776 p^{3} T^{5} + 1000056 p^{6} T^{6} + 680 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 73 | \( 1 + 54 T - 356153 T^{2} - 344801358 T^{3} + 80712341209 T^{4} + 132077540633940 T^{5} + 124229498711894494 T^{6} - 37768506737759197488 T^{7} - \)\(53\!\cdots\!66\)\( T^{8} - 37768506737759197488 p^{3} T^{9} + 124229498711894494 p^{6} T^{10} + 132077540633940 p^{9} T^{11} + 80712341209 p^{12} T^{12} - 344801358 p^{15} T^{13} - 356153 p^{18} T^{14} + 54 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 + 226 T - 487693 T^{2} - 344699266 T^{3} - 260271124975 T^{4} + 15628583326928 T^{5} + 22127295683141214 T^{6} + 48887116558883288988 T^{7} + \)\(10\!\cdots\!02\)\( T^{8} + 48887116558883288988 p^{3} T^{9} + 22127295683141214 p^{6} T^{10} + 15628583326928 p^{9} T^{11} - 260271124975 p^{12} T^{12} - 344699266 p^{15} T^{13} - 487693 p^{18} T^{14} + 226 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( ( 1 + 1568 T + 1587240 T^{2} + 1167350640 T^{3} + 850483913182 T^{4} + 1167350640 p^{3} T^{5} + 1587240 p^{6} T^{6} + 1568 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 89 | \( 1 + 1458 T + 542326 T^{2} + 83239488 T^{3} + 189980218918 T^{4} + 519794357813514 T^{5} + 567863209418477128 T^{6} - 52698436128215759790 T^{7} - \)\(39\!\cdots\!25\)\( T^{8} - 52698436128215759790 p^{3} T^{9} + 567863209418477128 p^{6} T^{10} + 519794357813514 p^{9} T^{11} + 189980218918 p^{12} T^{12} + 83239488 p^{15} T^{13} + 542326 p^{18} T^{14} + 1458 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( ( 1 + 2172 T + 2538830 T^{2} + 2100215688 T^{3} + 1807738972095 T^{4} + 2100215688 p^{3} T^{5} + 2538830 p^{6} T^{6} + 2172 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
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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
−4.49232786439808818563600522418, −4.48526243307980339055994426412, −4.20541517897927412430029832979, −4.18829946345064481011756437208, −4.01126813220167405180372543393, −3.86521482690341355801795773313, −3.69685241576617649493626795087, −3.67888285836591335788870902975, −3.47030558613885791974267163044, −3.12733297780365784659714703406, −2.98005502645857070240003176760, −2.96593244741680429921973144547, −2.83804410000250098266305236255, −2.71178653489864788094967212157, −2.46492186549737317835056369111, −2.40608580971813047655235889568, −1.90410383021030453911828330888, −1.68818502560730099704339451950, −1.48386248429162935478739138490, −1.46684967396321380591933634379, −1.21112973946991834012518566340, −1.10216170733093713601888398543, −0.60644053396057087768385755431, −0.17104929620653704466590756004, −0.04403570801587000310723219494, 0.04403570801587000310723219494, 0.17104929620653704466590756004, 0.60644053396057087768385755431, 1.10216170733093713601888398543, 1.21112973946991834012518566340, 1.46684967396321380591933634379, 1.48386248429162935478739138490, 1.68818502560730099704339451950, 1.90410383021030453911828330888, 2.40608580971813047655235889568, 2.46492186549737317835056369111, 2.71178653489864788094967212157, 2.83804410000250098266305236255, 2.96593244741680429921973144547, 2.98005502645857070240003176760, 3.12733297780365784659714703406, 3.47030558613885791974267163044, 3.67888285836591335788870902975, 3.69685241576617649493626795087, 3.86521482690341355801795773313, 4.01126813220167405180372543393, 4.18829946345064481011756437208, 4.20541517897927412430029832979, 4.48526243307980339055994426412, 4.49232786439808818563600522418
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.