Dirichlet series
| L(s) = 1 | − 4-s + 4·9-s − 16·11-s − 16-s + 6·19-s − 64·29-s − 18·31-s − 4·36-s + 16·41-s + 16·44-s + 21·49-s + 20·59-s + 40·61-s + 10·64-s + 88·71-s − 6·76-s + 16·79-s + 6·81-s − 20·89-s − 64·99-s − 28·101-s + 42·109-s + 64·116-s + 136·121-s + 18·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 4/3·9-s − 4.82·11-s − 1/4·16-s + 1.37·19-s − 11.8·29-s − 3.23·31-s − 2/3·36-s + 2.49·41-s + 2.41·44-s + 3·49-s + 2.60·59-s + 5.12·61-s + 5/4·64-s + 10.4·71-s − 0.688·76-s + 1.80·79-s + 2/3·81-s − 2.11·89-s − 6.43·99-s − 2.78·101-s + 4.02·109-s + 5.94·116-s + 12.3·121-s + 1.61·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 5^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.09876\times 10^{9}\) |
| Root analytic conductor: | \(2.04747\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.579201057\) |
| \(L(\frac12)\) | \(\approx\) | \(4.579201057\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 5 | \( 1 \) | |
| 7 | \( 1 - 3 p T^{2} + 47 p T^{4} - 474 p T^{6} + 27414 T^{8} - 474 p^{3} T^{10} + 47 p^{5} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16} \) | |
| good | 2 | \( 1 + T^{2} + p T^{4} - 7 T^{6} - 11 p T^{8} - 31 T^{10} - 15 T^{12} + 33 p^{2} T^{14} + 55 p^{2} T^{16} + 33 p^{4} T^{18} - 15 p^{4} T^{20} - 31 p^{6} T^{22} - 11 p^{9} T^{24} - 7 p^{10} T^{26} + p^{13} T^{28} + p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 + 8 T + 28 T^{2} + 120 T^{3} + 402 T^{4} + 36 p T^{5} + 384 T^{6} - 2384 T^{7} - 29421 T^{8} - 2384 p T^{9} + 384 p^{2} T^{10} + 36 p^{4} T^{11} + 402 p^{4} T^{12} + 120 p^{5} T^{13} + 28 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 17 T^{2} + 425 T^{4} - 6726 T^{6} + 86438 T^{8} - 6726 p^{2} T^{10} + 425 p^{4} T^{12} - 17 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 + 60 T^{2} + 1484 T^{4} + 20616 T^{6} + 278218 T^{8} + 400500 p T^{10} + 7285264 p T^{12} + 666359940 T^{14} - 6309930269 T^{16} + 666359940 p^{2} T^{18} + 7285264 p^{5} T^{20} + 400500 p^{7} T^{22} + 278218 p^{8} T^{24} + 20616 p^{10} T^{26} + 1484 p^{12} T^{28} + 60 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 3 T - 30 T^{2} - 125 T^{3} + 51 p T^{4} + 3616 T^{5} + 6406 T^{6} - 66486 T^{7} - 238148 T^{8} - 66486 p T^{9} + 6406 p^{2} T^{10} + 3616 p^{3} T^{11} + 51 p^{5} T^{12} - 125 p^{5} T^{13} - 30 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 44 T^{2} - 388 T^{4} - 24248 T^{6} + 734474 T^{8} + 14717524 T^{10} - 547791216 T^{12} - 1264808844 T^{14} + 414286345891 T^{16} - 1264808844 p^{2} T^{18} - 547791216 p^{4} T^{20} + 14717524 p^{6} T^{22} + 734474 p^{8} T^{24} - 24248 p^{10} T^{26} - 388 p^{12} T^{28} + 44 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 4 T + p T^{2} )^{16} \) | |
| 31 | \( ( 1 + 9 T + 13 T^{2} + 90 T^{3} + 749 T^{4} - 1629 T^{5} + 22230 T^{6} + 154161 T^{7} - 324118 T^{8} + 154161 p T^{9} + 22230 p^{2} T^{10} - 1629 p^{3} T^{11} + 749 p^{4} T^{12} + 90 p^{5} T^{13} + 13 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 108 T^{2} + 7374 T^{4} + 467096 T^{6} + 22703481 T^{8} + 873938840 T^{10} + 33932606830 T^{12} + 1175343473028 T^{14} + 38616591751924 T^{16} + 1175343473028 p^{2} T^{18} + 33932606830 p^{4} T^{20} + 873938840 p^{6} T^{22} + 22703481 p^{8} T^{24} + 467096 p^{10} T^{26} + 7374 p^{12} T^{28} + 108 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 256 p T^{5} + 104 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 43 | \( ( 1 - 5 p T^{2} + 21194 T^{4} - 1322001 T^{6} + 62692682 T^{8} - 1322001 p^{2} T^{10} + 21194 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 212 T^{2} + 20956 T^{4} + 1454040 T^{6} + 90802154 T^{8} + 5296739420 T^{10} + 289183381584 T^{12} + 15685077059564 T^{14} + 793076937068387 T^{16} + 15685077059564 p^{2} T^{18} + 289183381584 p^{4} T^{20} + 5296739420 p^{6} T^{22} + 90802154 p^{8} T^{24} + 1454040 p^{10} T^{26} + 20956 p^{12} T^{28} + 212 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 236 T^{2} + 25324 T^{4} + 1967208 T^{6} + 148062890 T^{8} + 10417342148 T^{10} + 635434140816 T^{12} + 36036607646708 T^{14} + 1967961208523651 T^{16} + 36036607646708 p^{2} T^{18} + 635434140816 p^{4} T^{20} + 10417342148 p^{6} T^{22} + 148062890 p^{8} T^{24} + 1967208 p^{10} T^{26} + 25324 p^{12} T^{28} + 236 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 10 T - 96 T^{2} + 1492 T^{3} + 4998 T^{4} - 116970 T^{5} + 158320 T^{6} + 3153670 T^{7} - 19452177 T^{8} + 3153670 p T^{9} + 158320 p^{2} T^{10} - 116970 p^{3} T^{11} + 4998 p^{4} T^{12} + 1492 p^{5} T^{13} - 96 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{8} \) | |
| 67 | \( 1 + 501 T^{2} + 138952 T^{4} + 26939913 T^{6} + 4029956509 T^{8} + 487571679936 T^{10} + 49074273490354 T^{12} + 4173403064575434 T^{14} + 302514422246172016 T^{16} + 4173403064575434 p^{2} T^{18} + 49074273490354 p^{4} T^{20} + 487571679936 p^{6} T^{22} + 4029956509 p^{8} T^{24} + 26939913 p^{10} T^{26} + 138952 p^{12} T^{28} + 501 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 1806 p T^{5} + 252 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 + 324 T^{2} + 50502 T^{4} + 5409800 T^{6} + 473911185 T^{8} + 35078829272 T^{10} + 2162596802518 T^{12} + 122333701830108 T^{14} + 7885309757584996 T^{16} + 122333701830108 p^{2} T^{18} + 2162596802518 p^{4} T^{20} + 35078829272 p^{6} T^{22} + 473911185 p^{8} T^{24} + 5409800 p^{10} T^{26} + 50502 p^{12} T^{28} + 324 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 8 T - 182 T^{2} + 1560 T^{3} + 19865 T^{4} - 148676 T^{5} - 1461174 T^{6} + 5521756 T^{7} + 104973092 T^{8} + 5521756 p T^{9} - 1461174 p^{2} T^{10} - 148676 p^{3} T^{11} + 19865 p^{4} T^{12} + 1560 p^{5} T^{13} - 182 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 436 T^{2} + 89684 T^{4} - 11804316 T^{6} + 1127629270 T^{8} - 11804316 p^{2} T^{10} + 89684 p^{4} T^{12} - 436 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 10 T - 260 T^{2} - 1572 T^{3} + 55478 T^{4} + 195874 T^{5} - 7285344 T^{6} - 5340058 T^{7} + 785732359 T^{8} - 5340058 p T^{9} - 7285344 p^{2} T^{10} + 195874 p^{3} T^{11} + 55478 p^{4} T^{12} - 1572 p^{5} T^{13} - 260 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 516 T^{2} + 120042 T^{4} - 17454416 T^{6} + 1892032563 T^{8} - 17454416 p^{2} T^{10} + 120042 p^{4} T^{12} - 516 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−3.03318099925028175991731550645, −2.73668341093799901774418811419, −2.72115410774127045642111610707, −2.63289590937365693647812815370, −2.57731617777496480113489871106, −2.46729201113551762688514491493, −2.33449134889661139999547826582, −2.21423578669554794080523883410, −2.19962865207576942532167656629, −2.10571859529667228071631049734, −2.05230848909953242746122392930, −1.95505052119967522859861589267, −1.91161374741507390954736468404, −1.90744400506504890934318377216, −1.79636395803195273539293131317, −1.77185203113840003604793935868, −1.63580945855839485378966123008, −1.42251439428817287457961989104, −0.914743042091760605761196334855, −0.893210635141111622154122924801, −0.73598118507422499887183694827, −0.67954874727708090024960517992, −0.59792853482908079345891294286, −0.36048736014284744019444323368, −0.32998435000809165745965217395, 0.32998435000809165745965217395, 0.36048736014284744019444323368, 0.59792853482908079345891294286, 0.67954874727708090024960517992, 0.73598118507422499887183694827, 0.893210635141111622154122924801, 0.914743042091760605761196334855, 1.42251439428817287457961989104, 1.63580945855839485378966123008, 1.77185203113840003604793935868, 1.79636395803195273539293131317, 1.90744400506504890934318377216, 1.91161374741507390954736468404, 1.95505052119967522859861589267, 2.05230848909953242746122392930, 2.10571859529667228071631049734, 2.19962865207576942532167656629, 2.21423578669554794080523883410, 2.33449134889661139999547826582, 2.46729201113551762688514491493, 2.57731617777496480113489871106, 2.63289590937365693647812815370, 2.72115410774127045642111610707, 2.73668341093799901774418811419, 3.03318099925028175991731550645
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.