Dirichlet series
| L(s) = 1 | − 7·9-s + 8·11-s − 6·19-s − 10·29-s − 40·31-s − 16·41-s + 36·49-s − 22·59-s + 24·61-s + 28·71-s − 26·79-s + 20·81-s − 10·89-s − 56·99-s − 16·101-s − 52·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 13·169-s + ⋯ |
| L(s) = 1 | − 7/3·9-s + 2.41·11-s − 1.37·19-s − 1.85·29-s − 7.18·31-s − 2.49·41-s + 36/7·49-s − 2.86·59-s + 3.07·61-s + 3.32·71-s − 2.92·79-s + 20/9·81-s − 1.05·89-s − 5.62·99-s − 1.59·101-s − 4.98·109-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-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 − 169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 5^{32} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.87936\times 10^{18}\) |
| Root analytic conductor: | \(3.89507\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 5^{32} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03617488161\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03617488161\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| 19 | \( ( 1 + 3 T - 43 T^{2} - 21 T^{3} + 1227 T^{4} - 21 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 3 | \( 1 + 7 T^{2} + 29 T^{4} + 4 p T^{6} - 292 T^{8} - 520 p T^{10} - 1576 T^{12} + 9019 T^{14} + 55153 T^{16} + 9019 p^{2} T^{18} - 1576 p^{4} T^{20} - 520 p^{7} T^{22} - 292 p^{8} T^{24} + 4 p^{11} T^{26} + 29 p^{12} T^{28} + 7 p^{14} T^{30} + p^{16} T^{32} \) |
| 7 | \( ( 1 - 18 T^{2} + 249 T^{4} - 2273 T^{6} + 2586 p T^{8} - 2273 p^{2} T^{10} + 249 p^{4} T^{12} - 18 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 11 | \( ( 1 - 2 T + 9 T^{2} - 45 T^{3} + 223 T^{4} - 45 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 13 | \( 1 + p T^{2} + 84 T^{4} - 4171 T^{6} - 69631 T^{8} - 436560 T^{10} + 2400958 T^{12} + 10061326 p T^{14} + 7365816 p^{2} T^{16} + 10061326 p^{3} T^{18} + 2400958 p^{4} T^{20} - 436560 p^{6} T^{22} - 69631 p^{8} T^{24} - 4171 p^{10} T^{26} + 84 p^{12} T^{28} + p^{15} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 + 67 T^{2} + 1913 T^{4} + 2344 p T^{6} + 892968 T^{8} + 16507324 T^{10} + 212184356 T^{12} + 3219557187 T^{14} + 61704982225 T^{16} + 3219557187 p^{2} T^{18} + 212184356 p^{4} T^{20} + 16507324 p^{6} T^{22} + 892968 p^{8} T^{24} + 2344 p^{11} T^{26} + 1913 p^{12} T^{28} + 67 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 106 T^{2} + 5775 T^{4} + 205208 T^{6} + 5179613 T^{8} + 93230502 T^{10} + 1059284623 T^{12} + 100222388 p T^{14} - 145747336167 T^{16} + 100222388 p^{3} T^{18} + 1059284623 p^{4} T^{20} + 93230502 p^{6} T^{22} + 5179613 p^{8} T^{24} + 205208 p^{10} T^{26} + 5775 p^{12} T^{28} + 106 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 5 T - 32 T^{2} + 51 T^{3} + 1313 T^{4} - 3448 T^{5} + 15591 T^{6} + 132895 T^{7} - 792020 T^{8} + 132895 p T^{9} + 15591 p^{2} T^{10} - 3448 p^{3} T^{11} + 1313 p^{4} T^{12} + 51 p^{5} T^{13} - 32 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 10 T + 95 T^{2} + 627 T^{3} + 3691 T^{4} + 627 p T^{5} + 95 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 37 | \( ( 1 - 54 T^{2} + 857 T^{4} - 80745 T^{6} + 5522262 T^{8} - 80745 p^{2} T^{10} + 857 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 8 T - 105 T^{2} - 550 T^{3} + 10439 T^{4} + 31486 T^{5} - 616215 T^{6} - 356064 T^{7} + 31448841 T^{8} - 356064 p T^{9} - 616215 p^{2} T^{10} + 31486 p^{3} T^{11} + 10439 p^{4} T^{12} - 550 p^{5} T^{13} - 105 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 + 231 T^{2} + 27597 T^{4} + 2272460 T^{6} + 146007348 T^{8} + 7898171744 T^{10} + 382114256080 T^{12} + 17344157658651 T^{14} + 757892525254801 T^{16} + 17344157658651 p^{2} T^{18} + 382114256080 p^{4} T^{20} + 7898171744 p^{6} T^{22} + 146007348 p^{8} T^{24} + 2272460 p^{10} T^{26} + 27597 p^{12} T^{28} + 231 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 + 70 T^{2} - 11 p T^{4} - 365860 T^{6} - 13976739 T^{8} + 426732730 T^{10} + 40627711703 T^{12} + 113264670720 T^{14} - 70566165833711 T^{16} + 113264670720 p^{2} T^{18} + 40627711703 p^{4} T^{20} + 426732730 p^{6} T^{22} - 13976739 p^{8} T^{24} - 365860 p^{10} T^{26} - 11 p^{13} T^{28} + 70 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 163 T^{2} + 8525 T^{4} + 1324 p T^{6} + 6571164 T^{8} + 2215264828 T^{10} + 153893709860 T^{12} + 3359790987759 T^{14} - 12107113557791 T^{16} + 3359790987759 p^{2} T^{18} + 153893709860 p^{4} T^{20} + 2215264828 p^{6} T^{22} + 6571164 p^{8} T^{24} + 1324 p^{11} T^{26} + 8525 p^{12} T^{28} + 163 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 11 T - 60 T^{2} - 979 T^{3} + 443 T^{4} + 4774 T^{5} - 391341 T^{6} + 1207965 T^{7} + 48891330 T^{8} + 1207965 p T^{9} - 391341 p^{2} T^{10} + 4774 p^{3} T^{11} + 443 p^{4} T^{12} - 979 p^{5} T^{13} - 60 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 12 T - 78 T^{2} + 856 T^{3} + 8493 T^{4} - 25976 T^{5} - 905558 T^{6} + 410364 T^{7} + 66133564 T^{8} + 410364 p T^{9} - 905558 p^{2} T^{10} - 25976 p^{3} T^{11} + 8493 p^{4} T^{12} + 856 p^{5} T^{13} - 78 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 288 T^{2} + 34980 T^{4} + 3392576 T^{6} + 388522410 T^{8} + 36346711712 T^{10} + 2614372052368 T^{12} + 204213886821984 T^{14} + 15609170671751923 T^{16} + 204213886821984 p^{2} T^{18} + 2614372052368 p^{4} T^{20} + 36346711712 p^{6} T^{22} + 388522410 p^{8} T^{24} + 3392576 p^{10} T^{26} + 34980 p^{12} T^{28} + 288 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 14 T - 87 T^{2} + 664 T^{3} + 19850 T^{4} - 38806 T^{5} - 1979478 T^{6} + 3533781 T^{7} + 107325432 T^{8} + 3533781 p T^{9} - 1979478 p^{2} T^{10} - 38806 p^{3} T^{11} + 19850 p^{4} T^{12} + 664 p^{5} T^{13} - 87 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 294 T^{2} + 44959 T^{4} + 4041228 T^{6} + 196484317 T^{8} - 509575266 T^{10} - 908279077625 T^{12} - 84929237407296 T^{14} - 6194219608054223 T^{16} - 84929237407296 p^{2} T^{18} - 908279077625 p^{4} T^{20} - 509575266 p^{6} T^{22} + 196484317 p^{8} T^{24} + 4041228 p^{10} T^{26} + 44959 p^{12} T^{28} + 294 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 13 T - 19 T^{2} - 94 T^{3} + 4641 T^{4} - 30533 T^{5} - 17014 T^{6} + 1995549 T^{7} - 18292886 T^{8} + 1995549 p T^{9} - 17014 p^{2} T^{10} - 30533 p^{3} T^{11} + 4641 p^{4} T^{12} - 94 p^{5} T^{13} - 19 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 475 T^{2} + 109580 T^{4} - 15844644 T^{6} + 1570393876 T^{8} - 15844644 p^{2} T^{10} + 109580 p^{4} T^{12} - 475 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 5 T - 209 T^{2} - 1332 T^{3} + 21515 T^{4} + 134213 T^{5} - 1599390 T^{6} - 5554751 T^{7} + 127743040 T^{8} - 5554751 p T^{9} - 1599390 p^{2} T^{10} + 134213 p^{3} T^{11} + 21515 p^{4} T^{12} - 1332 p^{5} T^{13} - 209 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 531 T^{2} + 138805 T^{4} + 27440820 T^{6} + 4773554764 T^{8} + 706952866932 T^{10} + 89695957751260 T^{12} + 10317391559984943 T^{14} + 1069090848073806961 T^{16} + 10317391559984943 p^{2} T^{18} + 89695957751260 p^{4} T^{20} + 706952866932 p^{6} T^{22} + 4773554764 p^{8} T^{24} + 27440820 p^{10} T^{26} + 138805 p^{12} T^{28} + 531 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.35776503904331014248180067209, −2.32643542069759297891727734339, −2.18406479684249371499446573633, −2.12652217394581903139870247448, −2.11713202619889180535601515894, −1.90486339338743646742087029061, −1.79933681564721986642235693263, −1.65889501031202581612545270873, −1.55634334099081512195129026729, −1.49433661455676854550494367740, −1.48618215502878343850189298177, −1.48053165795224767412359432621, −1.43867281638641617311354564227, −1.40585049020022463081376557475, −1.31620554946920880107835910674, −1.22497531527715149419935323470, −1.16765070659586012016622681824, −1.03771176357128095898027406578, −0.66502783248689151429665050911, −0.61529767362273450543460635692, −0.50505165056096775413731380216, −0.34350903886169666060919703724, −0.25542061804944408080424545035, −0.19603428420699741628802188549, −0.02332856883384785183001659625, 0.02332856883384785183001659625, 0.19603428420699741628802188549, 0.25542061804944408080424545035, 0.34350903886169666060919703724, 0.50505165056096775413731380216, 0.61529767362273450543460635692, 0.66502783248689151429665050911, 1.03771176357128095898027406578, 1.16765070659586012016622681824, 1.22497531527715149419935323470, 1.31620554946920880107835910674, 1.40585049020022463081376557475, 1.43867281638641617311354564227, 1.48053165795224767412359432621, 1.48618215502878343850189298177, 1.49433661455676854550494367740, 1.55634334099081512195129026729, 1.65889501031202581612545270873, 1.79933681564721986642235693263, 1.90486339338743646742087029061, 2.11713202619889180535601515894, 2.12652217394581903139870247448, 2.18406479684249371499446573633, 2.32643542069759297891727734339, 2.35776503904331014248180067209
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.