Dirichlet series
| L(s) = 1 | + 8·7-s − 12·19-s + 26·25-s − 24·31-s + 4·37-s − 8·43-s + 48·49-s + 28·67-s − 60·73-s + 32·79-s + 12·103-s + 44·109-s − 34·121-s + 127-s + 131-s − 96·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 208·175-s + 179-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 2.75·19-s + 26/5·25-s − 4.31·31-s + 0.657·37-s − 1.21·43-s + 48/7·49-s + 3.42·67-s − 7.02·73-s + 3.60·79-s + 1.18·103-s + 4.21·109-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s − 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 15.7·175-s + 0.0747·179-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.10314\times 10^{14}\) |
| Root analytic conductor: | \(2.83706\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.823168057\) |
| \(L(\frac12)\) | \(\approx\) | \(2.823168057\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 - 4 T + 4 T^{3} + 29 T^{4} + 4 p T^{5} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 5 | \( 1 - 26 T^{2} + 343 T^{4} - 3142 T^{6} + 22997 T^{8} - 145796 T^{10} + 840842 T^{12} - 4532272 T^{14} + 23186506 T^{16} - 4532272 p^{2} T^{18} + 840842 p^{4} T^{20} - 145796 p^{6} T^{22} + 22997 p^{8} T^{24} - 3142 p^{10} T^{26} + 343 p^{12} T^{28} - 26 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 + 34 T^{2} + 631 T^{4} + 7742 T^{6} + 5119 p T^{8} + 36244 T^{10} - 4803190 T^{12} - 80571856 T^{14} - 990277910 T^{16} - 80571856 p^{2} T^{18} - 4803190 p^{4} T^{20} + 36244 p^{6} T^{22} + 5119 p^{9} T^{24} + 7742 p^{10} T^{26} + 631 p^{12} T^{28} + 34 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 - 22 T^{2} + 597 T^{4} - 10874 T^{6} + 145448 T^{8} - 10874 p^{2} T^{10} + 597 p^{4} T^{12} - 22 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 84 T^{2} + 3336 T^{4} - 99240 T^{6} + 2736514 T^{8} - 66944172 T^{10} + 1418515200 T^{12} - 27661974396 T^{14} + 497038747923 T^{16} - 27661974396 p^{2} T^{18} + 1418515200 p^{4} T^{20} - 66944172 p^{6} T^{22} + 2736514 p^{8} T^{24} - 99240 p^{10} T^{26} + 3336 p^{12} T^{28} - 84 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 6 T + 41 T^{2} + 174 T^{3} + 567 T^{4} - 60 p T^{5} - 10940 T^{6} - 5748 p T^{7} - 514114 T^{8} - 5748 p^{2} T^{9} - 10940 p^{2} T^{10} - 60 p^{4} T^{11} + 567 p^{4} T^{12} + 174 p^{5} T^{13} + 41 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 84 T^{2} + 2840 T^{4} + 59304 T^{6} + 1536034 T^{8} + 2252628 p T^{10} + 1287398720 T^{12} + 22322505660 T^{14} + 403780671955 T^{16} + 22322505660 p^{2} T^{18} + 1287398720 p^{4} T^{20} + 2252628 p^{7} T^{22} + 1536034 p^{8} T^{24} + 59304 p^{10} T^{26} + 2840 p^{12} T^{28} + 84 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 82 T^{2} + 3369 T^{4} - 137570 T^{6} + 4900772 T^{8} - 137570 p^{2} T^{10} + 3369 p^{4} T^{12} - 82 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 4 T + 48 T^{2} + 404 T^{3} + 1502 T^{4} + 404 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 8 T + 72 T^{2} + 172 T^{3} + 1373 T^{4} + 172 p T^{5} + 72 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 2 T - 39 T^{2} + 686 T^{3} - 133 T^{4} - 26040 T^{5} + 186136 T^{6} + 542152 T^{7} - 8048970 T^{8} + 542152 p T^{9} + 186136 p^{2} T^{10} - 26040 p^{3} T^{11} - 133 p^{4} T^{12} + 686 p^{5} T^{13} - 39 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 204 T^{2} + 20760 T^{4} + 1374660 T^{6} + 65555246 T^{8} + 1374660 p^{2} T^{10} + 20760 p^{4} T^{12} + 204 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 2 T + 139 T^{2} + 134 T^{3} + 8158 T^{4} + 134 p T^{5} + 139 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 - 200 T^{2} + 20236 T^{4} - 1289200 T^{6} + 54604010 T^{8} - 1402967000 T^{10} + 1590108464 T^{12} + 2082887856200 T^{14} - 134280636932141 T^{16} + 2082887856200 p^{2} T^{18} + 1590108464 p^{4} T^{20} - 1402967000 p^{6} T^{22} + 54604010 p^{8} T^{24} - 1289200 p^{10} T^{26} + 20236 p^{12} T^{28} - 200 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 82 T^{2} + 1819 T^{4} - 293986 T^{6} - 25475671 T^{8} - 757240268 T^{10} + 15438451790 T^{12} + 2705625303848 T^{14} + 146204406396370 T^{16} + 2705625303848 p^{2} T^{18} + 15438451790 p^{4} T^{20} - 757240268 p^{6} T^{22} - 25475671 p^{8} T^{24} - 293986 p^{10} T^{26} + 1819 p^{12} T^{28} + 82 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 322 T^{2} + 52107 T^{4} - 6104798 T^{6} + 600323513 T^{8} - 51577574244 T^{10} + 3916097487742 T^{12} - 266299877019064 T^{14} + 16439030582633874 T^{16} - 266299877019064 p^{2} T^{18} + 3916097487742 p^{4} T^{20} - 51577574244 p^{6} T^{22} + 600323513 p^{8} T^{24} - 6104798 p^{10} T^{26} + 52107 p^{12} T^{28} - 322 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 70 T^{2} + 4306 T^{4} + 9720 T^{5} - 90560 T^{6} + 641520 T^{7} - 8009105 T^{8} + 641520 p T^{9} - 90560 p^{2} T^{10} + 9720 p^{3} T^{11} + 4306 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 14 T - 125 T^{2} + 1238 T^{3} + 431 p T^{4} - 158888 T^{5} - 2519254 T^{6} + 1335236 T^{7} + 243583474 T^{8} + 1335236 p T^{9} - 2519254 p^{2} T^{10} - 158888 p^{3} T^{11} + 431 p^{5} T^{12} + 1238 p^{5} T^{13} - 125 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 392 T^{2} + 75892 T^{4} - 9334040 T^{6} + 790096294 T^{8} - 9334040 p^{2} T^{10} + 75892 p^{4} T^{12} - 392 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 30 T + 591 T^{2} + 8730 T^{3} + 105765 T^{4} + 1168044 T^{5} + 12012906 T^{6} + 115920888 T^{7} + 1041481850 T^{8} + 115920888 p T^{9} + 12012906 p^{2} T^{10} + 1168044 p^{3} T^{11} + 105765 p^{4} T^{12} + 8730 p^{5} T^{13} + 591 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 16 T + 64 T^{2} - 56 T^{3} - 1765 T^{4} + 55364 T^{5} - 19696 T^{6} - 3181796 T^{7} + 14045608 T^{8} - 3181796 p T^{9} - 19696 p^{2} T^{10} + 55364 p^{3} T^{11} - 1765 p^{4} T^{12} - 56 p^{5} T^{13} + 64 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 438 T^{2} + 94557 T^{4} + 13226610 T^{6} + 1298783528 T^{8} + 13226610 p^{2} T^{10} + 94557 p^{4} T^{12} + 438 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 384 T^{2} + 64644 T^{4} - 8318208 T^{6} + 1128930634 T^{8} - 127988142720 T^{10} + 11460228290064 T^{12} - 1112004852903552 T^{14} + 109920677865950547 T^{16} - 1112004852903552 p^{2} T^{18} + 11460228290064 p^{4} T^{20} - 127988142720 p^{6} T^{22} + 1128930634 p^{8} T^{24} - 8318208 p^{10} T^{26} + 64644 p^{12} T^{28} - 384 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 234 T^{2} + 40305 T^{4} - 5045730 T^{6} + 535964996 T^{8} - 5045730 p^{2} T^{10} + 40305 p^{4} T^{12} - 234 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−2.49634489334945319146422458543, −2.42997797489056290230910278602, −2.34892887435399432528311611417, −2.29214188657053800574393688304, −2.25980349010269641437017619554, −2.25522239316860809256872679945, −2.13778319419448602122706233603, −2.07337608939352829597259235957, −1.96590051126470036150335558025, −1.96476523149036901443992185697, −1.64112758489543345953164979905, −1.62461765492225653114381305259, −1.56854812105395358604016997780, −1.45250412960641879429749869245, −1.43141586911129919552210958920, −1.37841145466071230861970633191, −1.27951021339037548021839135075, −1.03982138388953356249835519303, −0.987325216954501087233064342429, −0.917859136148403795076800645246, −0.847420042959029426363114093876, −0.62503383832755405301818974247, −0.50490843230729265885093622813, −0.20464630182711087734104956312, −0.10278192779873089782854603652, 0.10278192779873089782854603652, 0.20464630182711087734104956312, 0.50490843230729265885093622813, 0.62503383832755405301818974247, 0.847420042959029426363114093876, 0.917859136148403795076800645246, 0.987325216954501087233064342429, 1.03982138388953356249835519303, 1.27951021339037548021839135075, 1.37841145466071230861970633191, 1.43141586911129919552210958920, 1.45250412960641879429749869245, 1.56854812105395358604016997780, 1.62461765492225653114381305259, 1.64112758489543345953164979905, 1.96476523149036901443992185697, 1.96590051126470036150335558025, 2.07337608939352829597259235957, 2.13778319419448602122706233603, 2.25522239316860809256872679945, 2.25980349010269641437017619554, 2.29214188657053800574393688304, 2.34892887435399432528311611417, 2.42997797489056290230910278602, 2.49634489334945319146422458543
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.